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PrefaceThepresentbookisaboutdifferentialgeometryofspacecurvesandsurfaces.Theformulationandpresentationarelargelybasedonatensorcalculusapproach,whichisthedominanttrendinthemodernmathematicalliteratureofthissubject,ratherthanthegeometricapproachwhichisusuallyfoundinsomeoldstylebooks.Thebookisprepared,tosomeextent,aspartoftutorialsabouttopicsandapplicationsrelatedtotensorcalculus.Itcanthereforebeusedaspartofacourseontensorcalculusaswellasatextbookorareferenceforanintermediate-levelcourseondifferentialgeometryofcurvesandsurfaces.Apartfromgeneralbackgroundknowledgeinanumberofmathematicalbranchessuchascalculus,geometryandalgebra,animportantrequirementforthereaderanduserofthisbookisfamiliaritywiththeterminology,notationandconceptsoftensorcalculusatreasonablelevelsincemanyofthenotationsandconceptsofdifferentialgeometryinitsmodernstylearebasedontensorcalculus.Thebookcontainsamathematicalbackgroundsectioninthefirstchaptertooutlinesomeimportantpre-requiredmathematicalissues.However,thissectionisrestrictedtomaterialsrelateddirectlytothecontentsofdifferentialgeometryofthebookandhencethereaderandusershouldnotexpectthismathematicalbackgroundsectiontobecomprehensiveinanyway.Generalmathematicalknowledge,pluspossibleconsultationofmathematicaltextbooksrelatedtootherdisciplinesofmathematicswhenneeded,shouldthereforebeconsidered.Thebookisfurnishedwithanindexintheendofthebookaswellassetsofexercisesintheendofeachchaptertoprovideusefulrevisionsandpractice.Tofacilitatelinkingrelatedconceptsandparts,andhenceensurebetterunderstandingoftheprovidedmaterials,crossreferencingisusedextensivelythroughoutthebookwherethesereferralsarehyperlinkedintheelectronicversionofthebookfortheconvenienceoftheebookusers.Thebookalsocontainsaconsiderablenumberofgraphicillustrationstohelpthereadersanduserstovisualizetheideasandunderstandtheabstractconcepts.Thematerialsofdifferentialgeometryarestronglyinterlinkedandhenceanytextaboutthesubject,likethepresentone,willfacetheproblemofarrangingthematerialsinanaturalordertoensuregradualdevelopmentofconcepts.Inthisbookwelargelyfollowedsuchascheme.However,thisisnotalwayspossibleandhenceinsomecasesreferencesareprovidedtomaterialsinlaterpartsofthebookforconceptsneededinearlierparts.Nevertheless,inmostcasesbriefdefinitionsofthemainconceptsareprovidedinthefirstchapterinanticipationofmoredetaileddefinitionsandinvestigationsinthesubsequentchapters.Regardingthepreparationofthebook,everythingismadebytheauthorincludingallthegraphicillustrations,indexing,typesetting,bookcover,aswellasoveralldesign.Inthisregard,IshouldacknowledgetheuseofLaTeXtypesettingpackageandtheLaTeXbaseddocumentpreparationpackageLyXwhichfacilitatedthetypesettinganddesignofthebooksubstantially.TahaSochiLondon,March2017

TableofContentsPrefaceNomenclature1: Preliminaries1.1: DifferentialGeometry1.2: GeneralRemarks,ConventionsandNotations1.3: ClassifyingthePropertiesofCurvesandSurfaces1.3.1: LocalversusGlobalProperties1.3.2: IntrinsicversusExtrinsicProperties1.4: GeneralMathematicalBackground1.4.1: GeometryandTopology1.4.2: Functions1.4.3: Coordinates,TransformationsandMappings1.4.4: IntrinsicDistance1.4.5: BasisVectors1.4.6: FlatandCurvedSpaces

1.4.7: HomogeneousCoordinateSystems1.4.8: GeodesicCoordinates1.4.9: ChristoffelSymbolsforCurvesandSurfaces1.4.10: Riemann-ChristoffelCurvatureTensor1.4.11: RicciCurvatureTensorandScalar1.5: Exercises2: CurvesinSpace2.1: GeneralBackgroundaboutCurves2.2: MathematicalDescriptionofCurves2.3: CurvatureandTorsionofSpaceCurves2.3.1: Curvature2.3.2: Torsion2.4: GeodesicTorsion2.5: RelationshipbetweenCurveBasisVectorsandtheirDerivatives2.6: OsculatingCircleandSphere2.7: ParallelismandParallelPropagation2.8: Exercises3: SurfacesinSpace3.1: GeneralBackgroundaboutSurfaces3.2: MathematicalDescriptionofSurfaces3.3: SurfaceMetricTensor3.3.1: ArcLength3.3.2: SurfaceArea3.3.3: AngleBetweenTwoSurfaceCurves3.4: SurfaceCurvatureTensor3.5: FirstFundamentalForm3.6: SecondFundamentalForm3.6.1: DupinIndicatrix3.7: ThirdFundamentalForm3.8: FundamentalForms3.9: RelationshipbetweenSurfaceBasisVectorsandtheirDerivatives3.9.1: Codazzi-MainardiEquations3.10: SphereMapping3.11: GlobalSurfaceTheorems3.12: Exercises4: Curvature4.1: CurvatureVector4.2: NormalCurvature4.2.1: MeusnierTheorem4.3: GeodesicCurvature4.4: PrincipalCurvaturesandDirections4.5: GaussianCurvature4.6: MeanCurvature4.7: TheoremaEgregium4.8: Gauss-BonnetTheorem4.9: LocalShapeofSurface4.10: UmbilicalPoint4.11: Exercises5: SpecialCurves5.1: StraightLine5.2: PlaneCurve5.3: InvoluteandEvolute5.4: BertrandCurve5.5: SphericalIndicatrix5.6: SphericalCurve5.7: GeodesicCurve5.8: LineofCurvature5.9: AsymptoticLine5.10: ConjugateDirection5.11: Exercises6: SpecialSurfaces6.1: PlaneSurface6.2: QuadraticSurface

6.3: RuledSurface6.4: DevelopableSurface6.5: IsometricSurface6.6: TangentSurface6.7: MinimalSurface6.8: Exercises7: TensorDifferentiationoverSurfaces7.1: ExercisesReferencesAuthorNotes8: Footnotes

NomenclatureInthefollowingtable,wedefinesomeofthecommonsymbols,notationsandabbreviationswhichareusedinthebooktoprovideeasyaccesstothereader.∇ nabladifferentialoperator∇2 Laplacianoperator ~  isometricto,subscript partialderivativewithrespecttothefollowingindex(es);subscript covariantderivativewithrespecttothefollowingindex(es)1D,2D,3D,nD one-,two-,three-,n-dimensionaloverdot(e.g.ṙ) derivativewithrespecttogeneralparametertprime(e.g.r’) derivativewithrespecttonaturalparametersδ ⁄ δt absolutederivativewithrespecttot∂α,∂i partialderivativewithrespecttoαthandithvariablesa determinantofsurfacecovariantmetrictensora surfacecovariantmetrictensora11, a12, a22 coefficientsofsurfacecovariantmetrictensora11, a12, a22 coefficientsofsurfacecontravariantmetrictensoraαβ,aαβ,aβα surfacemetrictensororitscomponentsb determinantofsurfacecovariantcurvaturetensorb surfacecovariantcurvaturetensorB binormalunitvectorofspacecurveb11, b12, b22 coefficientsofsurfacecovariantcurvaturetensorbαβ,bαβ,bβα surfacecurvaturetensororitscomponentsC curveCB,CN,CT sphericalindicatricesofcurveCCe,Ci evoluteandinvolutecurvesCn ofclassncαβ,cαβ,cβα tensorofthirdfundamentalformoritscomponentsd Darbouxvectord1,d2 unitvectorsinDarbouxframedet determinantofmatrixds lengthofinfinitesimalelementofcurvedsB, dsN, dsT lengthoflineelementinbinormal,normal,tangentdirectionsdσ areaofinfinitesimalelementofsurfacee, f, g coefficientsofsecondfundamentalformE, F, G coefficientsoffirstfundamentalformℰ,ℱ,V numberofedges,facesandverticesofpolyhedronEi,Ej covariantandcontravariantspacebasisvectorsEα,Eβ covariantandcontravariantsurfacebasisvectorsEq./Eqs. Equation/Equationsf functionFig./Figs. Figure/Figuresg topologicalgenusofclosedsurfacegij,gij spacemetrictensororitscomponentsH meancurvatureIS,IIS,IIIS first,secondandthirdfundamentalforms

IS,IIS tensorsoffirstandsecondfundamentalforms

iff ifandonlyifJ JacobianoftransformationbetweentwocoordinatesystemsJ JacobianmatrixK GaussiancurvatureKt totalcurvatureL lengthofcurven normalunitvectortosurfaceN principalnormalunitvectortocurveP pointr,R radiusℛ Riccicurvaturescalarr positionvectorrα,rαβ 1stand2ndpartialderivativeofrwithsubscriptedvariablesR1,R2 principalradiiofcurvatureℝn n-dimensionalspace(usuallyEuclidean)Rij,Ri

j Riccicurvaturetensorof1stand2ndkindforspaceRαβ,Rα

β Riccicurvaturetensorof1stand2ndkindforsurfaceRijkl Riemann-Christoffelcurvaturetensorof1stkindforspaceRαβγδ Riemann-Christoffelcurvaturetensorof1stkindforsurfaceRi

jkl Riemann-Christoffelcurvaturetensorof2ndkindforspaceRα

βγδ Riemann-Christoffelcurvaturetensorof2ndkindforsurfaceRκ radiusofcurvatureRτ radiusoftorsions naturalparameterofcurverepresentingarclengthS surfaceST tangentsurfaceofspacecurvet generalparameterofcurveT functionperiodT tangentunitvectorofspacecurveTPS tangentspaceofsurfaceSatpointPtr traceofmatrixu geodesicnormalvectoru1, u2 surfacecoordinatesuα surfacecoordinateu, v surfacecoordinatesxi spacecoordinatexiα surfacebasisvectorinfulltensornotationx, y, z coordinatesin3Dspace(usuallyCartesian)[ij, k] Christoffelsymbolof1stkindforspace[αβ, γ] Christoffelsymbolof1stkindforsurfaceΓkij Christoffelsymbolof2ndkindforspaceΓγαβ Christoffelsymbolof2ndkindforsurfaceδij,δij,δji covariant,contravariantandmixedKroneckerdeltaδijkl generalizedKroneckerdeltaΔ discriminantofquadraticequation

ϵi1…in,ϵi1…in covariant,contravariantrelativepermutationtensorinnDspace

εi1…in,εi1…in covariant,contravariantabsolutepermutationtensorinnDspace

θ angleorparameter

θs sumofinterioranglesofpolygonκ curvatureofcurveκ1,κ2 principalcurvaturesofsurfaceatagivenpointκB,κT curvatureofbinormalandtangentsphericalindicatricesκg,κn geodesicandnormalcurvaturesκgu,κgv geodesiccurvaturesofuandvcoordinatecurvesκnu,κnv normalcurvaturesofuandvcoordinatecurvesK curvaturevectorKg,Kn geodesicandnormalcomponentsofcurvaturevectorλ directionparameterofsurfaceξ realparameterρ pseudo-radiusofpseudo-sphereρ, φ polarcoordinatesofplaneρ, φ, z cylindricalcoordinatesof3Dspaceσ areaofsurfacepatchτ torsionofcurveτB,τT torsionofbinormalandtangentsphericalindicatricesτg geodesictorsionφ angleorparameterχ Eulercharacteristicω realparameterNote:duetotherestrictionsontheavailabilityandvisibilityofsymbolsinthemobiformat,aswellassimilarformattingissues,weshoulddrawtheattentionoftheebookreaderstothefollowingpoints:1.Barsoversymbols,whichareusedintheprintedversion,werereplacedbytildes.However,forconveniencewekeptusingtheterms“barred”and“unbarred”inthetexttorefertothesymbolswithandwithouttildes.2.Thesquarerootsymbolinmobiis√()wheretheargumentiscontainedinsidetheparentheses.Forexample,thesquarerootofaissymbolizedas√(a).3.Inthemobiformat,superscriptsareautomaticallydisplayedbeforesubscriptsunlesscertainmeasuresaretakentoforcetheoppositewhichmaydistortthelookofthesymbolandmaynotevenbetherequiredformatwhenthesuperscriptsandsubscriptsshouldbesidebysidewhichisnotpossibleinthemobitextandliveequations.Therefore,forconvenienceandaestheticreasonsweonlyforcedtherequiredorderofthesubscriptsandsuperscriptsorusedimagedsymbolswhenitisnecessary;otherwiseweleftthesymbolstobedisplayedaccordingtothemobichoicealthoughthismaynotbeideallikedisplayingtheChristoffelsymbolofthesecondkindas:ΓγαβorthegeneralizedKroneckerdeltaas:δαδβωinsteadoftheirnormallook

as: and .

Chapter1PreliminariesInthischapter,weprovidepreliminarymaterialsintheformofageneralintroductionaboutdifferentialgeometry,remarksabouttheconventionsandnotationsusedinthisbook,classificationofthepropertiesofcurvesandsurfaces,andageneralmathematicalbackgroundrelatedtodifferentialgeometryofcurvesandsurfaces.

1.1 DifferentialGeometryDifferentialgeometryisabranchofmathematicsthatlargelyemploysmethodsandtechniquesofotherbranchesofmathematicssuchasdifferentialandintegralcalculus,topologyandtensoranalysistoinvestigategeometricalissuesrelatedtoabstractobjects,suchasspacecurvesandsurfaces,andtheirpropertieswheretheseinvestigationsaremostlyfocusedonthesepropertiesatsmallscales.Theinvestigationsofdifferentialgeometryalsoincludecharacterizingcategoriesoftheseobjects.Thereisalsoacloselinkbetweendifferentialgeometryandthedisciplinesofdifferentialtopologyanddifferentialequations.Differentialgeometrymaybecontrastedwith“algebraicgeometry”whichisanotherbranchofgeometrythatusesalgebraictoolstoinvestigategeometricissuesmainlyofglobalnature.Theinvestigationofthepropertiesofcurvesandsurfacesindifferentialgeometryarecloselylinked.Forinstance,investigatingthecharacteristicsofspacecurvesisextensivelyexploitedintheinvestigationofsurfacessincecommonpropertiesofsurfacesaredefinedandquantifiedintermsofthepropertiesofcurvesembeddedinthesesurfaces.Forexample,severalaspectsofthesurfacecurvatureatapointaredefinedandquantifiedintermsoftheparametersofthesurfacecurvespassingthroughthatpoint.

1.2 GeneralRemarks,ConventionsandNotationsFirst,weshouldremarkthatthepresentbookislargelybasedoninvestigatingcurvesandsurfacesembeddedina3DflatspacecoordinatedbyarectangularCartesiansystem.Inmostcases,“surface”and“space”inthepresentbookmean2Dand3Dmanifoldsrespectively.Anotherremarkisthattwistedcurvescanresideina2Dmanifold(surface)orinahigherdimensionalitymanifold(usually3Dspace).Henceweusuallyuse“surfacecurves”and“spacecurves”torefertothetypeofthemanifoldofresidence.However,inmostcasesasinglecurvecanbeviewedasresidentofmorethanonemanifoldandhenceitisasurfaceandspacecurveatthesametime.Forexample,acurveembeddedinasurfacewhichinitsturnisembeddedina3Dspaceisasurfacecurveandaspacecurveatthesametime.Consequently,inthisbookthesetermsshouldbeinterpretedflexibly.Manystatementsformulatedintermsofaparticulartypeofmanifoldcanbecorrectlyandeasilyextendedtoanothertypewithminimaladjustmentsofdimensionalityandsymbolism.Moreover,“space”insomestatementsshouldbeunderstoodinitsgeneralmeaningasamanifoldembracingthecurvenotasoppositeto“surface”andhenceitcanincludea2Dspace,i.e.surface.Followingtheconventionofseveralauthors,whendiscussingissuesrelatedto2Dand3DmanifoldstheGreekindicesrangeover1, 2whiletheLatinindicesrangeover1, 2, 3.Therefore,theuseofGreekandLatinindicesshouldingeneralindicatethetypeoftheintendedmanifoldunlessitisstatedotherwise.WeuseuindexedwithsuperscriptGreekletters(e.g.uαanduβ)tosymbolizesurfacecoordinates(seeFootnote1in§8↓)whilewelargelyusexindexedwithsuperscriptLatinletters(e.g.xiandxj)torepresentspaceCartesiancoordinatesalthoughtheyaresometimesusedtorepresentspacegeneralcurvilinearcoordinates.Commentsareusuallyaddedtoclarifythesituationifnecessary.ArelatedissueisthattheindexedEaremostlyusedforthesurface,ratherthanthespace,basisvectorswheretheyaresubscriptedorsuperscriptedwithGreekindices,e.g.EαandEβ.However,insomecasesindexedEareusedforspacebasisvectors;inwhichcasetheyaredistinguishedbyusingLatinindices,e.g.EiandEj.Whenthebasisvectorsareindexednumericallyratherthansymbolically,thedistinctionbetweensurfaceandspacebasesshouldbeobviousfromthecontextifitisnotstatedexplicitly.RegardingtheChristoffelsymbolsofthefirstandsecondkindofvariousmanifolds,theymaybebasedonthespacemetricorthesurfacemetric.Hence,whenanumberofChristoffelsymbolsinacertaincontextorequationarebasedonmorethanonemetric,thetypeofindices,i.e.GreekorLatin,canbeusedasanindicatortotheunderlyingmetricwheretheGreekindicesrepresentthesurface(e.g.[αβ, γ]andΓγ

αβ)whiletheLatinindicesrepresentthespace(e.g.[ij, k]andΓk

ij).Nevertheless,commentsaregenerallyaddedtoaccountforpotentialoversight.Inparticular,whentheChristoffelsymbolsarenumbered(e.g.Γ1

22),insteadofbeingindexedsymbolically,commentswillbeaddedtoclarifythesituation.Forbrevity,convenienceandcleannotationincertaincontexts,weuseanoverdot(e.g.ṙ)toindicatederivativewithrespecttoageneralparametertwhileweuseaprime(e.g.r’)toindicatederivativewithrespecttoanaturalparametersrepresentingarclength.Forthesamereasons,subscriptsarealsousedoccasionallytosymbolizepartialderivativewithrespecttothesubscriptvariables,e.g.fvandrαβ.Weshouldalsoremarkthatwefollowthesummationconvention,whichislargelyusedintensorcalculus.Commentsareaddedinafewexceptionalcaseswherethisconventiondoesnotapply.Wedeliberatelyuseavarietyofnotationsforthesameconceptsforthepurposeofconvenienceandtofamiliarizethereaderwithdifferentnotationsallofwhichareincommonuseintheliteratureofdifferentialgeometryandtensorcalculus.Havingproficiencyinthesesubjectsrequiresfamiliaritywiththesevarious,andsometimesconflicting,notations.Moreover,insomesituationstheuseofoneofthesenotationsortheotheriseithernecessaryoradvantageousdependingonthelocationandcontext.Animportantexampleofusingdifferentnotationsisthatrelatedtosurfacecoordinateswhereweusebothu, vandu1, u2torepresentthesecoordinatessinceeachoneofthesenotationshasadvantagesovertheotherdependingonthecontext;moreoverthelatterisnecessaryforexpressingtheequationsofdifferentialgeometryinindicialtensorforms.Anotherimportantexamplerelatedtotheuseofavarietyofnotationsistheemploymentofdifferentsymbolsforthecoefficientsofthefirstandsecondfundamentalforms(i.e.E, F, G, e, f, g)ononehandandthecoefficientsofthesurfacecovariantmetrictensorandthe

surfacecovariantcurvaturetensor(i.e.a11, a12, a22, b11, b12, b22)ontheotherdespitetheequivalenceofthesecoefficients,thatis(E, F, G, e, f, g) = (a11, a12, a22, b11, b12, b22),andhencethesedifferentnotationscanbereplacedbyjustone.However,wekeepbothnotationsastheyarebothincommonuseintheliteratureofdifferentialgeometryandtensorcalculus;moreoverinmanysituationstheuseofoneofthesenotationsinparticulariseithernecessaryoradvantageous,asindicatedearlier.Weshouldalsomentionthatweuseboldfacesymbols(e.g.A)tomarktensorsofrank > 0(includingmatriceswhichrepresentsuchtensors)intheirsymbolicnotation,whileweusenormalfacesymbols(e.g.c)forlabelingscalarsaswellastensorsofrank > 0andtheircomponentsintheirindicialformwhereinthelattercasethesymbolsareindexed(e.g.Aiandbβα).Also,squarebracketscontainingarraysareusedtorepresentmatriceswhileverticalbarsareusedtorepresentdeterminants.

1.3 ClassifyingthePropertiesofCurvesandSurfacesTherearetwomainclassificationsforthepropertiesofcurvesandsurfacesembeddedinhigherdimensionalityspaces;theseclassificationsarelocalpropertiesversusglobalproperties,andintrinsicpropertiesversusextrinsicproperties.Inthefollowingsubsectionswebrieflyinvestigatetheseoverlappingclassifications.

1.3.1 LocalversusGlobalProperties

Thepropertiesofcurvesandsurfacesmaybecategorizedintotwomaingroups:localandglobalwherethesepropertiesdescribethegeometryofthecurvesandsurfacesinthesmallandinthelargerespectively.Thelocalpropertiescorrespondtothecharacteristicsoftheobjectintheimmediateneighborhoodofapointontheobjectsuchasthecurvatureofacurveorsurfaceatthatpoint,whiletheglobalpropertiescorrespondtothecharacteristicsoftheobjectonalargescaleandoverextendedpartsoftheobjectsuchasthenumberofstationarypointsofacurveorasurfaceorbeingaone-sidesurfacelikeMobiusstrip(Fig.1↓)whichislocallyadouble-sidesurface.

Figure1 Mobiusstrip.Asindicatedearlier,differentialgeometryofspacecurvesandsurfacesismainlyconcernedwiththelocalproperties.Theinvestigationofglobalpropertiesnormallyinvolvetopologicaltreatmentswhicharebeyondthecommontoolsandmethodsthatareusuallyemployedindifferentialgeometry.Infact,thereisaspecialbranchofdifferentialgeometrydedicatedtotheinvestigationofglobal(orinthelarge)properties.Regardingthepresentbook,itismostlylimitedtodifferentialgeometryinthesmallalthoughanumberofglobaldifferentialgeometricissuesareinvestigatedfollowingthetraditionpursuedinthecommontextbooksofdifferentialgeometry.

1.3.2 IntrinsicversusExtrinsicProperties

Anotherclassificationofthepropertiesofcurvesandsurfaces,whichisbasedontheirrelationtotheembeddingexternalspacewhichtheyresidein,maybemadewherethepropertiesaredividedintointrinsicandextrinsic.ThefirstcategorycorrespondstothosepropertieswhichareindependentintheirexistenceanddefinitionfromtheambientspacewhichembracestheobjectsuchasthedistancealongagivencurveortheGaussiancurvatureofasurfaceatagivenpoint(see§4.5↓),whilethesecondcategoryisrelatedtothosepropertieswhichdependintheirexistenceanddefinitionontheexternalembeddingspacesuchashavinganormalvectoratapointonthecurveorthesurface.Moretechnically,theintrinsicproperties(seeFootnote2in§8↓)aredefinedandexpressedintermsofthemetrictensorwhichisformulatedindifferentialgeometryasthefirstfundamentalform(see§3.5↓)whiletheextrinsicpropertiesareexpressedintermsofthesurfacecurvaturetensorwhichisformulatedindifferentialgeometryasthesecondfundamentalform(see§3.6↓).Aswewillseeinseveralplacesofthisbook,somequantitiescanbeexpressedonceintermsofthecoefficientsofthefirstfundamentalformexclusivelyandonceintermsofexpressionsinvolvingthecoefficientsofthesecondfundamentalformaswell.Inthisregard,aquantityisclassifiedasintrinsicifitcanbeexpressedasafunctionofthecoefficientsofthefirstfundamentalformonlyevenifitcanalsobeexpressedintermsinvolvingthecoefficientsofthesecondfundamentalform.Weshouldalsoremarkthatextrinsicpropertiesarenormallyexpressedintermsofbothformsalthoughthesepropertiesarecharacterizedbybeingexpressednecessarilyintermsofthesecondformsincethisisthefeaturethatdistinguishesthemfromintrinsicproperties.Theideaofintrinsicandextrinsicpropertiesmaybeillustratedbyaninhabitantofasurfacewitha2Dperception(hereafterthiscreaturewillbecalled“2Dinhabitant”)wherehecandetectandmeasureintrinsicpropertiesbutnotextrinsicpropertiesastheformerdonotrequireappealingtoanexternalembedding3Dspaceinwhichthesurfaceisimmersedwhilethelatterdo.Hence,insimpletermsallthepropertiesthatcanbedetectedandmeasuredbya2Dinhabitantareintrinsictothesurfacewhilealltheotherpropertiesareextrinsic.A1Dinhabitantofacurvemayalsobeused,toalesserextent,analogouslytodistinguishbetweenintrinsicandextrinsicpropertiesofsurfaceandspacecurves(referforexampleto§2.3↓).Theso-called“intrinsicgeometry”ofthesurfacecomprisesthecollectionofalltheintrinsicpropertiesofthesurface.Whentwosurfacescanhaveacoordinatesystemoneachsuchthatthefirstfundamentalformsofthetwosurfacesareidenticalateachpairofcorrespondingpointsonthetwosurfacesthenthetwosurfaceshaveidenticalintrinsicgeometry.Suchsurfacesareisometric(see§6.5↓)andcanbemappedoneachotherbyatransformationthatpreservesthecurvelengths,theanglesandthesurfaceareas.Moreover,ifthesesurfacesaresubjectedtoidenticalcoordinatetransformations,theyremainidenticalintheirintrinsicproperties.

1.4 GeneralMathematicalBackgroundInthissection,weprovideageneralmathematicalbackgroundoutliningconceptsanddefinitionsneededingeneraltounderstandtheforthcomingmaterialsofdifferentialgeometry.However,itshouldberemarkedthatsomeofthematerialsprovidedinthepresentsectionwhicharerelatedtoconceptsfromothersubjectsofmathematics,suchascalculusandtopology,areelementarybecauseofthelimitsonthetextsize;moreoverthebookisnotpreparedasatextaboutthesesubjects.Thepurposeoftheseoutlinesanddefinitionsistoprovideabasicunderstandingoftherelatedideas.Thereadersare,therefore,advisedtorefertotextbooksonthosesubjectsformoretechnicalandextensiverevelations.Thissectionalsocontainssomematerialsthatmaynotbeneededintheforthcomingchaptersbuttheyareusuallydiscussedandinvestigatedindifferentialgeometrytexts.Ourobjectiveofincludingsuchmaterialsisforthepresentbooktobemorecomprehensiveandcompatiblewithsimilardifferentialgeometrytexts.

1.4.1 GeometryandTopology

Here,wedefinesomegeometricshapeswhichareusedasexamplesandprototypesintheforthcomingsectionsandchaptersandtheymaynotbefamiliartosomereaders.Commongeometricshapes(likestraightline,circleandsphere)arecommonknowledgeatthislevelandhencetheywillnotbedefined.Wealsointroducesomebasictopologicalconceptswhichwillbeneededintheforthcomingpartsofthebook.AsurfaceofrevolutionisanaxiallysymmetricsurfacegeneratedbyaplanecurveC(see§5.2↓)revolvingaroundastraightlineLcontainedintheplaneofthecurvebutnotintersectingthecurve.ThecurveCiscalledtheprofileofthesurfaceandthelineLiscalledtheaxisofrevolutionwhichisalsotheaxisofsymmetryofthesurface.Meridiansofasurfaceofrevolutionareplanecurvesonthesurfaceformedbytheintersectionofaplanecontainingtheaxisofrevolutionwiththesurface,andhencethemeridiansareidenticalversionsoftheprofilecurveC.Parallelsofasurfaceofrevolutionarecirclesgeneratedbyintersectingthesurfacebyplanesperpendiculartotheaxisofrevolution,andhencetheyrepresentthepathsofspecificpointsontheprofilecurveC.Forspheres,thesecurvesarecalledmeridiansoflongitudeandparallelsoflatitude.Onanysurfaceofrevolution,meridiansandparallelsintersectatrightangles.Asurfaceofrevolutionmaybegeneratedinitssimplestformbyrevolvingacurvex = f(z)aroundthez-axisofaCartesiancoordinatesystem.Thesurfacecanthenbeparameterizedas:(1)x = fcosφ(2)y = fsinφ(3)z = zwhere(f, φ, z)representthestandardcylindricalcoordinates(ρ, φ, z).Thehelix(Fig.2↓)isaspacecurvecharacterizedbyhavingatangentvectorthatformsaconstantanglewithaspecifieddirectionwhichisthedirectiondefinedbyitsaxisofrotation.Thehelixcanbedefinedparametricallyby:(4)x = acos(θ)(5)y = asin(θ)(6)z = bθwherea, barenon-zerorealconstantsandθisarealparameterwith − ∞ < θ <  + ∞.Thecirclemayberegardedasadegenerateformofhelixcorrespondingtob = 0,whileastraightlinemayberegardedasanotherdegenerateformofhelixcorrespondingtoa = 0.

Figure2 Helixanditsparameters.Thetorus(Fig.3↓)isasurfaceofrevolutionwhoseprofilecurveCisacircle.Itcanbedefinedparametricallyby:(7)x = (R + rcosφ)cosθ(8)y = (R + rcosφ)sinθ(9)z = rsinφwhereRisthetorusradius(i.e.thedistancebetweenthecenterofthegeneratingcircleandthecenterofsymmetryofthetoruswhichistheperpendicularprojectionofthecirclecenterontheaxisofrevolution),ristheradiusofthegeneratingcircle(r < R),φ ∈ [0, 2π)istheangleofvariationofr,andθ ∈ [0, 2π)istheangleofvariationofR.

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Figure3 Torusanditsparameters.Theellipsoid(Fig.4↓)isaquadraticsurface(see§6.2↓)thatcanbedefinedparametricallyby:(10)x = asinθcosφ(11)y = bsinθsinφ(12)z = ccosθwherea, b, carenon-zerorealconstantsandθ, φarerealparameterswith0 ≤ θ ≤ πand0 ≤ φ < 2π.

Figure4 Ellipsoid.

Thehyperboloidofonesheet(Fig.5↓)isaquadraticsurfacethatcanbedefinedparametricallyby:(13)x = acoshξcosθ(14)y = bcoshξsinθ(15)z = csinhξwherea, b, carenon-zerorealconstantsandξ, θarerealparameterswith − ∞ < ξ <  + ∞and0 ≤ θ < 2π.

Figure5 Hyperboloidofonesheet.Thehyperboloidoftwosheets(Fig.6↓)isaquadraticsurfacethatcanbedefinedparametricallyby:(16)x = asinhξcosθ(17)y = bsinhξsinθ(18)z = ccoshξwherea, b, carenon-zerorealconstantsandξ, θarerealparameterswith0 ≤ ξ < ∞and0 ≤ θ < 2π.

Figure6 Hyperboloidoftwosheets.Theellipticparaboloid(Fig.7↓)isaquadraticsurfacethatcanbedefinedparametricallyby:(19)x = a√(ξ)cosθ(20)y = b√(ξ)sinθ(21)z = cξwherea, b, carenon-zerorealconstantsandξ, θarerealparameterswith0 ≤ ξ < ∞and0 ≤ θ < 2π.

Figure7 Ellipticparaboloid.Thehyperbolicparaboloid(Fig.8↓)isaquadraticsurfacethatcanbedefinedparametricallyby:(22)x = aξ(23)y = bω(24)z = cξωwherea, b, carenon-zerorealconstantsandξ, ωarerealparameterswith − ∞ < ξ <  + ∞and − ∞ < ω <  + ∞.

Figure8 Hyperbolicparaboloid.Theparaboliccylinder(Fig.9↓)isaquadraticsurfacegeneratedinitssimplestformbytranslatingaparabola(expressedinyasaquadraticfunctionofx,ortheotherwayaround,inacanonicalform)alongthez-direction.Hence,itcanbeparameterizedbythefollowingequations:(25)x = ξ(26)y = aξ2(27)z = bωwherea, barenon-zerorealconstantsandξ, ωarerealparameterswith − ∞ < ξ <  + ∞and − ∞ < ω <  + ∞.

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Figure9 Paraboliccylinder.Thecatenaryisaplanecurve(see§5.2↓)withahyperboliccosineshape.Itcanbedefinedparametricallyby:(28)x = acosh(ξ ⁄ a)(29)z = ξwhereaisanon-zerorealconstantand − ∞ < ξ <  + ∞isarealparameter.Thecatenoid(Fig.10↓)isasurfaceofrevolutiongeneratedbyrevolvingacatenaryarounditsdirectrix,whichisthez-axisintheaboveformulation,andhenceitcanbedefinedparametricallyby:(30)x = acosh(ξ ⁄ a)cosθ(31)y = acosh(ξ ⁄ a)sinθ(32)z = ξwhereaisanon-zerorealconstantandξ, θarerealparameterswith − ∞ < ξ <  + ∞and0 ≤ θ < 2π.

Figure10 Catenoid.Thehelicoid(Fig.11↓)isaruledsurface(see§6.3↓)withthepropertythatforeachpointPonthesurfacethereisahelixpassingthroughPandcontainedentirelyinthesurface.Itcanbedefinedparametricallyby:(33)x = aξcosθ(34)y = aξsinθ(35)z = bθwherea, barenon-zerorealconstantsandξ, θarerealparameterswith − ∞ ≤ ξ ≤  + ∞and − ∞ ≤ θ <  + ∞.

Figure11 Helicoid.Themonkeysaddle(Fig.12↓)isasaddlesurfacethatcanbedefinedparametricallyby:(36)x = ξ(37)y = ω(38)z = ξ3 − 3ξω2

whereξ, ωarerealparameterswith − ∞ < ξ <  + ∞and − ∞ < ω <  + ∞.

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Figure12 Monkeysaddle.Theenneper(Fig.13↓)isaself-intersectingsurfacethatcanbedefinedparametricallyby:(39)x =  − (ξ3 ⁄ 3) + ξ + ξω2

(40)y =  − ξ2ω − ω + (ω3 ⁄ 3)(41)z = ξ2 − ω2

whereξ, ωarerealparameterswith − ∞ < ξ <  + ∞and − ∞ < ω <  + ∞.

Figure13 Enneper.Thetractrix(Fig.14↓)isaplanecurvestarting(seeFootnote3in§8↓)fromthepoint(ρ, 0)onthex-axiswiththepropertythatthelengthofthelinesegmentofitstangentbetweenthetangencypointandthepointofintersectionwiththez-axisisequaltoρwhereρisarealconstant(ρ > 0).Hence,

thetractrixisasolutionofthefollowingdifferentialequation:(42)dz ⁄ dx = ±√(ρ2 − x2) ⁄ xwhere0 < x ≤ ρandwiththeconditionz(ρ) = 0.Theplussigninthisequationcorrespondstothelowerpart(z < 0)whiletheminussigncorrespondstotheupperpart.TheBeltramipseudo-sphere(Fig.14↓)isasurfaceofrevolutiongeneratedbyrevolvingatractrixarounditsasymptotewhichisthez-axisintheaboveformulation.Thepseudo-spherecanbedefinedparametricallyby:(43)x = asinθcosφ(44)y = asinθsinφ(45)z = a[cosθ + ln(tan(θ ⁄ 2))]whereaisarealconstantandθ, φarerealparameterswith0 < θ < πand0 ≤ φ < 2π.Theconstantρofthetractrixwhichgeneratesthepseudo-sphereiscalledthepseudo-radiusofthepseudo-sphere.

Figure14 Tractrix(leftframe)andpseudo-sphere(rightframe).Itshouldberemarkedthatthegeometricshapesthatwedefinedparametricallyinthissubsection(e.g.torus,ellipsoid,andhyperbolicparaboloid)canalsobedefinedbyotherparametricformsaswellasbyexplicitCartesianandnon-Cartesianforms(seee.g.§6.2↓).TheEulercharacteristic,orEuler-Poincarecharacteristic,isatopologicalparameterofclosedsurfaces(see§3.1↓)which,forpolyhedralsurfaces,isgivenby:(46)χ = V + ℱ − ℰwhereχistheEulercharacteristicofthesurface,andV, ℱ, ℰarethenumbersofvertices,facesandedgesofthepolyhedron.ExamplesofEulercharacteristicofsomecommonpolyhedronsaregiveninFig.15↓.

Figure15 Polyhedralsurfaces:tetrahedron(leftframe),cube(middleframe)andoctahedron(rightframe).TheEulercharacteristicforthesesurfacesisgivenrespectivelyby:χ = 4 + 4 − 6,χ = 8 + 6 − 12,andχ = 6 + 8 − 12,whichinallcasesisequalto2.TheEulercharacteristiccanalsobedefinedformoregeneraltypesofsurface.TheEulercharacteristicofacompactorientable(see§3.1↓)non-polyhedralsurface,likesphereandtorus,canbeobtainedbypolygonaldecompositionbasedondividingtheentiresurfaceintoafinitenumberofnon-overlappingcurvilinearpolygonswhichshareatmostedgesorvertices(seeFig.16↓).Theaboveformula(Eq.46↑)isthenused,asforpolyhedralsurfaces,todeterminetheEulercharacteristicofthesurface.

Figure16 Polygonaldecompositionofasphereintofournon-overlappingcurvilinearpolygons.Insimpleterms,thetopologicalgenusofasurfaceisthenumberofhandlesortopologicalholesonthesurface.Forexample,theellipsoid(Fig.4↑)isasurfaceofgenus0whilethetorus(Fig.3↑)isasurfaceofgenus1.Similarly,thesurfacesinFig.17↓areofgenus2and3.ForanorientablesurfaceofgenusgtheEulercharacteristicχisrelatedtothegenusby:(47)χ = 2(1 −  g)

Figure17 Examplesofsurfacesofgenus2(leftframe)andgenus3(rightframe).ThelengthofastraightlinesegmentconnectingtwopointsinaEuclideanspaceisameasureofthedistance(initscommonsense)betweenthetwopoints(seeFootnote4in§8↓).Thelengthofapolygonalarc(Fig.18↓a)isthesumofthelengthsofitsstraightsegments.Thelengthofanarcofageneralizedtwistedspacecurveisthelimitofthelengthofanasymptoticpolygonalarc(Fig.18↓b)asthelengthofthelongeststraightlinesegmentoftheasymptoticpolygonalarctendstozero.

Figure18 (a)Polygonalarcand(b)twistedcurve(solid)withtwoofitsasymptoticpolygonalarcs(dashedanddotted)wherethedashedrepresentsabetterapproximationtothelengthofthetwistedcurvethanthedotted.Theareaofapolygonalplanefragment(e.g.triangleorsquare)isameasureofitstwo-dimensionalexpansion(seeFootnote5in§8↓).Theareaofasurfacepatchconsistingentirelyofpolygonalplanefragmentsisthesumoftheareasofitspolygonalplanefragments.Theareaofapatchofageneralizedtwistedspacesurfaceisthelimitoftheareaofitsasymptoticpolygonalpatchastheareaofthelargestofitspolygonalfragmentstendstozero(refertoFig.19↓).

Figure19 Asmoothsurface(bottom)approximatedbyacoarseasymptoticpolygonalsurface(top)andafineasymptoticpolygonalsurface(middle)wherethefinerepresentsabetterapproximationtotheareaofthesmoothsurfacethanthecoarse.

1.4.2 Functions

Thedomainofafunctionalmapping:f:ℝm → ℝnisthelargestsetofℝmonwhichthemappingisdefined.Abicontinuousfunctionormappingisacontinuousfunctionwithacontinuousinverse.AscalarfunctionisofclassCnifthefunctionandallofitsfirstn(butnotn + 1)partialderivativesdoexistandarecontinuous.Avectorfunction(e.g.apositionvectorrepresentingaspacecurveorsurface)isofclassCnifoneofitscomponentsisofthisclasswhilealltheothercomponentsareofthisclassorhigher.AcurveorasurfaceisofclassCnifitismathematicallyrepresentedbyafunctionofthisclass.InthiscontextweshouldremarkthatinmostcasesthepurposeofimposingtheconditionofhavingafunctionofclassCnistohaveafunctionwhichisatleastofthisclassandhencetheconditionismetbyhavingafunctionofthisclassorhigher.Infact,thisisgenerallythecaseinthisbookwhenthisconditionisimposedunlessthereisanobviousindicatorthatthefunctionisstrictlyofthisclass.AnotherremarkisthatthereshouldbenoconfusionbetweenthebareCsymbolandthesuperscriptedCsymbol,likeC2,asthebareCisusuallyusedinthisbooktosymbolizeacurvewhilethesuperscriptedCstandsfortheabovedifferentiabilitycondition.Ingrossterms,a“smooth”or“sufficientlysmooth”curveorsurfacemeansthatthefunctionalrelationthatrepresentstheobjectissufficientlydifferentiablefortheintendedobjective,beingofclassCnatleastwherenistheminimumrequirementforthedifferentiabilityindextosatisfytherequiredconditions.AdeletedneighborhoodofapointPona1Dintervalonthereallineisdefinedasthesetofallpointsx ∈ ℝintheintervalsuchthat:(48)0 < |x − xP| < ϵwherexPisthecoordinateofPonthereallineandϵisapositiverealnumber.Hence,thedeletedneighborhoodincludesallthepointsintheopeninterval(xP − ϵ, xP + ϵ)excludingxPitself.Foraspacecurve(whichisnotstraightingeneral)representedbyr = r(t),whereristhespatialrepresentationofthecurveandtisageneralparameterinthecurverepresentation,thedefinitionappliestotheneighborhoodoftPwheretPisthevalueoftcorrespondingtothepointPonthecurve.DeletedneighborhoodofatwistedcurvemayalsobeidentifiedbybeingconfinedinacircleofradiusϵcenteredatP,howeverthisappliesonlytoplanecurves.Therefore,itmaybeidentifiedbybeingconfinedinasphereofradiusϵcenteredatPtobemoregeneral.AdeletedneighborhoodofapointPona2Dflatsurfaceisdefinedasthesetofallpoints(x, y) ∈ ℝ2onthesurfacesuchthat:(49)0 < √([x − xP]2 + [y − yP]2) < ϵwhere(xP, yP)arethecoordinatesofPontheplaneandϵisapositiverealnumber.Hence,thedeletedneighborhoodincludesallthepointsinsideacirculardiscofradiusϵandcenter(xP, yP)excludingthecenteritself.Foraspacesurface(whichisnotflatingeneral)representedbyr = r(u, v),whereristhespatialrepresentationofthesurfaceandu, varethesurfacecoordinatesontheuvplane(see§1.4.3↓)thatmaponthesurface,thedefinitionappliestotheneighborhoodof(uP, vP)where(uP, vP)arethecoordinatesonthe2DuvplanecorrespondingtothepointPonthesurface.Similartotwistedcurves,deletedneighborhoodofatwistedsurfacemayalsobeidentifiedbybeingconfinedinasphereofradiusϵcenteredatP.Theabovedefinitionsofdeletedneighborhoodsofcurvesandsurfacescanbeeasilyextendedtospacesofhigherdimensionality.Forinstance,ina3Dspacetheneighborhoodiscontainedinasphericalsurfacewithradiusϵ.However,thisisnotneededinthisbookwhosefocusisspacecurvesandsurfaces.Aquadraticexpression:(50)Q(x, y) = a1x2 + 2a2xy + a3y2ofrealcoefficientsa1, a2, a3andrealvariablesx, yisdescribedas“positivedefinite”ifitpossessespositivevalues( > 0)forallpairs(x, y) ≠ (0, 0).ThesufficientandnecessaryconditionforQtobepositivedefiniteisthat:

(51)a1 > 0and(a1a3 − a2a2) > 0Thesetwoconditionsnecessitateathirdconditionthatis:a3 > 0sincethesecoefficientsarereal.ThefirstvariationofafunctionalFmaybedefinedbytheGateauxderivativeofthefunctionalas:(52)

wherexandharevariablefunctionsandξisarealscalarparameter.TheEuler-LagrangevariationalprincipleisamathematicalrulewhoseobjectiveistominimizeormaximizeacertainfunctionalF(f)whichdependsonafunctionf.ItisrepresentedmathematicallybyapartialdifferentialequationwhosesolutionsoptimizetheparticularfunctionalF.TheEuler-Lagrangeprincipleinitsgeneric,simpleandmostcommonformisgivenmathematicallyby:(53)

wheref(x, y, yx)isafunctionofthegivenvariablesthatoptimizesthefunctionalF,yisafunctionofx,andyxisthederivativeofywithrespecttox.

1.4.3 Coordinates,TransformationsandMappingsTheJacobianJofatransformationbetweentwocoordinatesystems,labeledasunbarredandbarred,isthedeterminantoftheJacobianmatrixJofthetransformationbetweenthesesystems,thatis:(54)

wheretheindexedxandxarethecoordinatesintheunbarredandbarredcoordinatesystemsinannDspace.Anadmissiblecoordinatetransformationmaybedefinedgenericallyasamappingrepresentedbyasufficientlydifferentiablesetofequationsanditisinvertiblebyhavinganon-vanishingJacobian(J ≠ 0)(seeFootnote6in§8↓).Aninvariantpropertyofacurveorasurfaceisapropertywhichisindependentofadmissiblecoordinatetransformationsandparameterizations.Itshouldbenotedthataninvariantpropertymaybeinvariantwithrespecttocertaintypesoftransformationorparameterizationbutnotwithrespecttoothersandhenceitissometimesusedgenericallywherethecontextshouldbetakenintoconsiderationforsensibleinterpretation.Anorthogonalcoordinatetransformationisacombinationoftranslation,rotationandreflectionofaxes.TheJacobianoforthogonaltransformationsisunity,thatisJ = ±1.TheorthogonaltransformationisdescribedaspositiveiffJ =  + 1andnegativeiffJ =  − 1.Positiveorthogonaltransformationsconsistsolelyoftranslationandrotation(possiblytrivialonesasinthecaseoftheidentitytransformation)whilenegativeorthogonaltransformations

includereflection,byapplyinganoddnumberofaxesreversal,aswell.Positivetransformationscanbedecomposedintoaninfinitenumberofcontinuouslyvaryinginfinitesimalpositivetransformationseachoneofwhichimitatesanidentitytransformation.Suchadecompositionisnotpossibleinthecaseofnegativeorthogonaltransformationsbecausetheshiftfromtheidentitytransformationtoreflectionisimpossiblebyacontinuousprocess.AsurfaceSina3Dspacemaybedescribeddirectlybythethreespatialcoordinatesofagiven3Dcoordinatesystem(e.g.x, y, zofaCartesiansystem)orbythetwosurfacecoordinates(e.g.u, v).Intheformercase,thedescriptionisratherfamiliarwhereeachpointonthespacesurfaceisidentifieddirectlybythreespatialcoordinateslinkedbyagivenfunctionalrelation(see§3.1↓and6.2↓).Inthelattercase,a2DdomainoverwhichthesurfaceSisdefinedisintroduced,wherethisdomainisidentifiedbytwoindependentvariables(usuallyu, voru1, u2).Thedomainisusuallyassumed,forsimplicity,tobeaplane(whichmaybecalledtheuvplaneorparametersplane)whereagridofuandvcurvesaredefinedoverthisplane.Again,forsimplicitythecurvesintheseuandvfamiliesareusuallydefinedasstraightlines;moreover,thecurvesineachfamilyareregularlyspacedwiththetwofamiliesofuandvcurvesbeingmutuallyorthogonal(seeFig.20↓).Althoughtheseassumptionsaboutthedomainandaboutthegridandparametercurvescanbeviolated(e.g.byusinga2Dpolarorcurvilinearcoordinatesystemovertheparametersplaneinsteadoftheabove-described2DrectangularCartesian),inmostcasesthereisnoadvantageindoingso.Alongeachcurveoftheuandvfamiliesonlyonevariable(eitheruorv)varieswhiletheothervariable(voru)isheldconstant.So,alonganycurveoftheufamilyvisconstantandalonganycurveofthevfamilyuisconstant.

Figure20 Theuvparametersplanecorrespondingtoaspacesurfaceandthemappingoftheuvgrid.ThesurfaceSisthendefinedbyafunctionalmappingfromtheuvplaneontothespacesurfaceS.Thisfunctionalmappingidentifiesanypointonthesurfaceinthe3Dspacecorrespondingtoanypointinitsdomainintheuvplane.Hence,theuvgridontheuvplaneisprojectedbythisfunctionalmappingonacorrespondinguvgridonthesurfaceS.Asweareassumingthatthesurfaceisembeddedina3Dspace,thefunctionalmappingconsistsofthreeindependentrelationswhereeachrelationcorrelatestheuandvcoordinatesofagivenpointinthedomaintooneofthethreespatialcoordinates(x1, x2, x3)ofacorrespondingpointofthesurfaceintheembeddingspace,thatis:x1(u, v),x2(u, v)andx3(u, v).Whiletheuvgridontheuvplaneisusuallyaplanar,orthogonalandregularlyspacedgrid,thecorrespondinggridonthesurfaceSisnotnecessarilysobecauseitisgenerallya3Dgridthatfollowsthebendsandvariationsofthetwistedspacesurface.Inthiscontext,“coordinatecurves”(whicharealsocalledparametriccurvesorparametriclines)onasurfacearedefinedasthemaporprojectionoftheuvcurvesoftheuvgridoftheparametersplaneontothespacesurface,asdescribedabove,andhencetheyarecurvesalongwhichonlyonecoordinatevariable(uorv)varieswhiletheothercoordinatevariable(voru)remainsconstant.So,alongtheucoordinatecurvesvisheldconstantwhilealongthevcoordinatecurvesuisheldconstant(seeFig.20↑).Itisnoteworthythatalthoughthesurfacecoordinatesu, v(oru1, u2)primarilyrepresentthecoordinatesontheuvparametersplanewhichmapsonthesurface,intheliteratureofdifferentialgeometrytheyaresometimesused(toeasethenotation)aslabelsforthecurvesonthesurface.Hence,vigilanceisrequiredtoavoidconfusion.AregularrepresentationofclassCm(m > 0)ofasurfacepatchSina3DEuclideanspaceisdefinedasafunctionalmappingofanopensetΩintheuvplaneontoSthatsatisfiesthefollowingtwoconditions:

1.ThefunctionalmappingrelationisofclassCmovertheentireΩ.2.TheJacobianmatrixofthetransformationbetweentherepresentationofthesurfaceinthe3Dspaceandits2Ddomainisofrank2forallthepointsinΩ.WeremarkthatforafunctionalmappingoftheformS(u, v) = (S1(u, v), S2(u, v), S3(u, v)),theaforementionedJacobianmatrixisgivenby:(55)

Infact,havingaJacobianmatrixofrank2forthetransformation,accordingtotheabovecondition,isequivalenttotheconditionthatE1 ×  E2 ≠  0whereE1 = ∂urandE2 = ∂vrarethesurfacebasisvectors(see§1.4.5↓,3.1↓and3.2↓),whicharethetangentstotheuandvcoordinatecurvesrespectively,andr = r(u, v)isthe3Dspatialrepresentationofthecurves.HavingaJacobianmatrixofrank2isalsoequivalenttohavingawell-definedtangentplanetothesurfaceattherelatedpoint.Inthisregard,wenotethat“rank”herereferstoitsmeaninginlinearalgebraandshouldnotbeconfusedwiththerankofatensorwhichisrelatedtothenumberofitsfreeindicesalthoughthereisaconnectionbetweenthetwo.Aswewillsee,aregularpointonasurfaceisapointthatsatisfiestheaboveconditionaboutthebasisvectors.Apointonasurfacewhichisnotregulariscalledsingular.Singularityoccurseitherbecauseofageometricreason,whichisthecaseforinstancefortheapexofacone,orbecauseoftheparticularparametricrepresentationofthesurface.Whilethefirsttypeofsingularityisinherentandhenceitcannotberemoved,thesecondtypecanberemovedbychangingtherepresentation.Correspondingpointsontwocurvesrefertotwopoints,oneoneachcurve,withacommonvalueofacommonparameterofthetwocurves.Whenthetwocurveshavetwodifferentparameterizationsthenaone-to-onecorrespondencebetweenthetwoparametersshouldbeestablishedandthecorrespondingpointsthenrefertotwopointswithcorrespondingvaluesofthetwoparameters.Correspondingpointsontwosurfacescanbedefinedinasimilarmannertakingintoaccountthatsurfacesrequiretwoindependentparametersintheiridentification.Inmanycasesoftheoreticalandpracticalsignificance,amixedtensorAiα,whichiscontravariantwithrespecttotransformationsinspacecoordinatesxiandcovariantwithrespecttotransformationsinsurfacecoordinatesuα,maybedefined.Followingacoordinatetransformationinwhichboththespaceandsurfacecoordinateschange,thetensorAiαwillbegiveninthenew(barred)systemby:(56)Ãiα = Ajβ(∂xi ⁄ ∂xj)(∂uβ ⁄ ∂ũα)Moregenerally,tensorswithspaceandsurfacecontravariantindicesandspaceandsurfacecovariantindices(e.g.Aiαjβ)canalsobedefinedinasimilarmanner.Theextensionoftheabovetransformationruletoincludesuchtensorscanbeeasilyachievedbyfollowingtheobviouspatternseeninthelastequation.

1.4.4 IntrinsicDistanceTheintrinsicdistancebetweentwopointsonasurfaceisthegreatestlowerbound(orinfimum)ofthelengthsofallregulararcsconnectingthetwopointsonthesurface.Theintrinsicdistanceisanintrinsicpropertyofthesurface.Theintrinsicdistancedbetweentwopointsisinvariantunderalocalisometricmapping(see§3.1↓),thatis:(57)d(f(P1), f(P2)) = d(P1, P2)wherefisanisometricmappingfromasurfaceS1toasurfaceS2(see§3.1↓and6.5↓),P1andP2arethetwopointsonS1andf(P1)andf(P2)aretheirimagesonS2.Infact,thismaybetakenasthedefinitionofisometricmapping,i.e.itisthemappingthatpreservesintrinsicdistance.ThefollowingconditionsapplytotheintrinsicdistancedbetweenpointsP1,P2andP3:1.Symmetry:d(P1, P2) = d(P2, P1).2.Triangleinequality:d(P1, P3) ≤ d(P1, P2) + d(P2, P3).3.Positivedefiniteness:d(P1, P2) > 0withd(P1, P2) = 0iffP1andP2arethesamepoint.AnarcCconnectingtwopoints,P1andP2,onasurfaceisdescribedasanarcofminimumlengthbetweenP1andP2ifthelengthofCisequaltotheintrinsicdistancebetweenP1andP2.Theexistenceanduniquenessofanarcofminimumlengthbetweentwospecificpointsonasurfaceisnotguaranteed,i.e.itmaynotexitandifitdoesexistitmaynotbeunique(referto§5.7↓forexamples).Yes,forcertaintypesofsurfacesuchanarcdoesexistanditisunique.Forexample,onasimplyconnected(see§3.1↓)planesurfacethereexistsanarcofminimumlengthbetweenanytwopointsontheplaneanditisunique;thisarcisthestraightlinesegmentconnectingthetwopoints.

1.4.5 BasisVectors

Thesetofbasisvectorsinagivenmanifoldplaysapivotalroleinthetheoreticalconstructionofthegeometryofthemanifold,andthisappliestothebasisvectorsindifferentialgeometrywherethesevectorsareusedinthedefinitionandconstructionofessentialconceptsandobjectssuchasthemetrictensorofthesurface.Thesetofbasisvectorsmayalsobeemployedtoserveasamovingcoordinateframefortheenvelopingspaceoftheirunderlyingconstructions(see§2.2↓and§3.2↓).Thedifferentialgeometryofcurvesandsurfacesemploystwomainsetsofbasisvectors:1.Onesetisconstructedonspacecurvesandconsistsofthreeunitvectors:thetangentT,thenormalNandthebinormalBtothecurve.2.Anothersetisconstructedonsurfacesandconsistsoftwolinearlyindependentvectorswhicharethetangentstothecoordinatecurvesofthesurface,E1 = (∂ r ⁄ ∂u1)andE2 = (∂ r ⁄ ∂u2),plustheunitnormalvectortothesurfacen,wherer(u1, u2)isthespatialrepresentationofasurfacecoordinatecurve,andu1andu2arethesurfacecoordinatesasdiscussedearlierandaswillbeinvestigatedfurtherlaterinthebook(referto§1.4.3↑and3.2↓).Eachoneoftheabovebasissetsisdefinedoneachregularpointofthecurveorthesurfaceandhenceingeneralthevectorsineachoneofthesebasissetsvaryfromonepointtoanother,i.e.theyarepositiondependent.ThevectorsE1andE2,whicharenotnecessarilyofunitlength,varyinmagnitudeanddirectionwhiletheunitvectors(whicharetherest)varyindirection.Also,whilethevectorsoftheT, N, Bsetaremutuallyorthogonal,thevectorsoftheE1,  E2,  nsetisnotnecessarilysosinceE1andE2arenotorthogonalingeneralalthoughnisorthogonaltobothE1andE2.Thesurfacebasisvectors,E1andE2,aregiveninfulltensornotationby(∂xi ⁄ ∂uα)(i = 1, 2, 3andα = 1, 2)whichisusuallyabbreviatedasxiα.Thesevectorscanbeseenascontravariantspacevectorsorascovariantsurfacevectors.Furtherdetailsaboutthiswillbegivenin§3.3↓.Weremarkthatothersetsofbasisvectorsarealsodefinedandemployedindifferentialgeometry,aswewillseeinthefuture(referto§4.1↓and4.4↓).

1.4.6 FlatandCurvedSpaces

Amanifold,suchasa2Dsurfaceora3Dspace,iscalled“flat”ifitispossibletofindacoordinatesystemforthemanifoldwithadiagonalmetrictensor

whosealldiagonalelementsare±1;thespaceiscalled“curved”otherwise.Moreformally,annDspaceisdescribedasaflatspaceiffitispossibletofindacoordinatesystemforwhichthelineelementdsisgivenby:(58)(ds)2 = ζ1(dx1)2 + ζ2(dx2)2 + … + ζn(dxn)2 = Σni = 1ζi(dxi)2wheretheindexedζare±1whiletheindexedxarethecoordinatesofthespace.Forthespacetobeflat(i.e.globallynotjustlocally),theconditiongivenbyEq.58↑shouldapplyalloverthespaceandnotjustatcertainpointsorregions.Anexampleofflatspaceisthe3DEuclideanspacewhichcanbecoordinatedbyarectangularCartesiansystemwhosemetrictensorisdiagonalwithallthediagonalelementsbeing + 1.Anotherexampleisthe4DMinkowskispace-timemanifoldwhosemetricisdiagonalwithelementsof±1.All1DspacesareEuclideanandhencetheycannotbecurvedintrinsically,sotwistedcurvesarecurvedonlywhenviewedexternallyfromtheembeddingspacewhichtheyresidein,e.g.the2Dspaceofasurfacecurveorthe3Dspaceofaspacecurve.Anexampleofcurvedspaceisthe2Dsurfaceofasphereoranellipsoid.AnecessaryandsufficientconditionforannDspacetobeintrinsicallyflatisthattheRiemann-Christoffelcurvaturetensor(see§1.4.10↓)ofthespacevanishesidentically.Hence,cylindersareintrinsicallyflat,sincetheirRiemann-Christoffelcurvaturetensorvanishesidentically,althoughtheyarecurvedasseenextrinsicallyfromtheembedding3Dspace.Ontheotherhand,planesareintrinsicallyandextrinsicallyflat.Inbrief,aspaceisintrinsicallyflatifftheRiemann-Christoffelcurvaturetensorvanishesidenticallyoverthespace,anditisextrinsically(aswellasintrinsically)flatiffthecurvaturetensor(see§3.4↓)vanishesidenticallyoverthespace.ThisisbecausetheRiemann-Christoffelcurvaturetensorcharacterizesthespacecurvaturefromanintrinsicperspectivewhilethecurvaturetensorcharacterizesthespacecurvaturefromanextrinsicperspective.DuetothestrongconnectionbetweentheGaussiancurvature(see§4.5↓)andtheRiemann-Christoffelcurvaturetensorwhichimpliesthateachoneofthesewillvanishiftheotherdoes(seeforexampleEq.92↓),weseethathavinganidenticallyvanishingGaussiancurvatureisanothersufficientandnecessaryconditionfora2Dspacetobeflat.ItshouldberemarkedthattheGaussiancurvatureindifferentialgeometryisdefinedfor2DspacesalthoughtheconceptmaybeextendedtohigherdimensionalitymanifoldsintheformofRiemanniancurvature.Thisissuewillbediscussedfurtherlaterinthebook(see§4.4↓).Acurvedspacemayhaveconstantnon-vanishingcurvaturealloverthespace,orhavevariablecurvatureandhencethecurvatureispositiondependent.AnexampleofaspaceofconstantcurvatureisthesurfaceofasphereofradiusRwhosecurvature(i.e.Gaussiancurvature)is(1 ⁄ R2)ateachpointofthesurface.Torus(Fig.3↑),ellipsoid(Fig.4↑)andparaboloids(Figs.7↑and8↑)aresimpleexamplesofsurfaceswithvariablecurvature.Aswewillsee,therearevariouscharacterizationsandquantificationsforthecurvature;henceinthepresentcontext“curvature”maybeagenericterm.For2Dsurfaces,curvatureusuallyreferstotheGaussiancurvature(see§4.5↓)whichisstronglylinkedtotheRiemanniancurvature,asindicatedabove.SchurtheoremrelatedtonDspaces(n > 2)ofconstantcurvaturestatesthat:iftheRiemann-Christoffelcurvaturetensorateachpointofaspaceisafunctionofthecoordinatesonly,thenthecurvatureisconstantalloverthespace.Schurtheoremmayalsobestatedas:theRiemanniancurvatureisconstantoveranisotropicregionofannD(n > 2)Riemannianspace.Thefocusofthebook,however,islimitedtospacesoflowerdimensionality.ThegeometryofcurvedspacesisusuallydescribedastheRiemanniangeometry.OneapproachforinvestigatingtheRiemanniangeometryofacurvedmanifoldistoembedthemanifoldinaEuclideanspaceofhigherdimensionalityandinspectthepropertiesofthemanifoldfromthisperspective.Thisapproachislargelyfollowedinthepresentbookwherethegeometryofcurved2Dspaces(twistedsurfaces)isinvestigatedbyimmersingthesurfacesina3DEuclideanspaceandexaminingtheirpropertiesasviewedfromthisexternalenveloping3Dspace.Suchanexternalviewisnecessaryforexaminingtheextrinsicgeometryofthesurfacebutnotitsintrinsicgeometry.Asimilarapproachisalsofollowedintheinvestigationofsurfaceandspacecurves.Asurfacewithpositive/negativeGaussiancurvature(see§4.5↓)ateachpointisdescribedasasurfaceofpositive/negativecurvature.Ellipsoids(Fig.4↑),hyperboloidsoftwosheets(Fig.6↑)andellipticparaboloids(Fig.7↑)areexamplesofsurfacesofpositivecurvaturewhilehyperboloidsofonesheet(Fig.5↑)andhyperbolicparaboloids(Fig.8↑)areexamplesofsurfacesofnegativecurvature.Asurfacewithpositive/negativeGaussiancurvatureateachpointmayalsobedescribedasasurfaceofconstantcurvaturesinceitssignisconstantalloverthesurfacealthoughitsmagnitudemaybevariable.Ingeneral,atwistedsurfacepossessescoordinate-dependentcurvaturewhichvariesinmagnitudeandsignandmaytakeallpossiblesigns(i.e. < 0,0and > 0)overdifferentpointsorregions.Thegeometricdescriptionandquantificationofflatspacesaresimplerthanthoseofcurvedspaces,andhenceingeneralthedifferentialgeometryofflatspacesismoremotivatingandlesschallengingthanthatofcurvedspaces.However,asthereisasubjectiveelementinthistypeofstatements,itmaynotapplytoeveryone.

1.4.7 HomogeneousCoordinateSystemsWhenallthediagonalelementsofadiagonalmetrictensorofaflatspaceare + 1,thecoordinatesystemisdescribedashomogeneous.InthiscasethelineelementdsofEq.58↑becomes:(59)(ds)2 = dxidxiAnexampleofhomogeneouscoordinatesystemsistheorthonormalCartesiansystem(x, y, z)ofa3DEuclideanspace(Fig.21↓).Ahomogeneouscoordinatesystemcanbetransformedtoanotherhomogeneouscoordinatesystemonlybylineartransformations.Anycoordinatesystemobtainedfromahomogeneouscoordinatesystembyanorthogonaltransformationisalsohomogeneous.Asaconsequenceofthelaststatements,infinitelymanyhomogeneouscoordinatesystemscanbeconstructedinanyflatspace.

Figure21 OrthonormalCartesiancoordinatesystemanditsbasisvectorse1,  e2ande3ina3Dspace.Acoordinatesystemofaflatspacecanalwaysbehomogenizedbyallowingthecoordinatestobeimaginary.Thisisdonebyredefiningthecoordinatesas:(60)Xi = √(ζi)xi

wherethenewcoordinatesXiareimaginarywhenζi =  − 1.Consequently,thelineelementwillbegivenby:(61)(ds)2 = dXidXiwhichisofthesameformasEq.59↑.AnexampleofahomogeneouscoordinatesystemwithsomerealandsomeimaginarycoordinatesisthecoordinatesystemofaMinkowski4Dspace-timerelatedtothemechanicsofLorentztransformations.

1.4.8 GeodesicCoordinatesItisalwayspossibletointroducecoordinatesatparticularpointsinamulti-dimensionalmanifoldsothattheChristoffelsymbols(see§1.4.9↓)vanishatthesepoints.Thesecoordinatesarecalledgeodesiccoordinates.Geodesiccoordinatesareemployedaslocalcoordinatesystemsmainlyforthepurposeofachievingcertainadvantages,aswillbeoutlinednext.ThesegeodesicsystemsfortheseparticularpointsarealsodescribedaslocallyCartesiancoordinates.TheallocatedpointsatwhichtheChristoffelsymbolsaremadetovanishinthesecoordinatesaredescribedasthepoles.Basedontheaboveandwhatwewillseelaterinthebook,geodesiccoordinatesofspacesurfacesarenormallytakenastheequivalentoftheCartesiancoordinatesofaflatspace.Hence,non-geodesiccoordinateson2DspacesmaybecomparedtogeneralcurvilinearcoordinatesingeneralnDspaces.Themainreasonfortheuseofgeodesiccoordinatesisthatthecovariantandabsolutederivatives(see§7↓)insuchsystemsbecomerespectivelypartialandtotalderivativesatthepolessincetheChristoffelsymboltermsinthecovariantandabsolutederivativeexpressionsvanishatthesepoints.Anytensorpropertycanthenbeeasilyprovedinthegeodesicsystematthepoleandconsequentlygeneralizedtoothersystemsduetotheinvarianceofthezerotensorunderpermissiblecoordinatetransformations.Iftheallocatedpoleisageneralpointinthespace,thepropertyisthenestablishedoverthewholespace.InanyRiemannianspaceitisalwayspossibletofindacoordinatesystemforwhichthecoordinatesaregeodesicateverypointofagivenanalyticcurve.Moreover,thereisaninfinitenumberofwaysbywhichgeodesiccoordinatescanbedefinedoveracoordinatepatch(see§3.1↓).Weshouldremarkthatsomeauthorsdefinegeodesiccoordinatesonacoordinatepatchofasurfaceasacoordinatesystemwhoseuandvcoordinatecurvefamiliesareorthogonal,withoneofthesefamilies(uorv)beingafamilyofgeodesiccurves(referto§5.7↓).Hence,“geodesiccoordinates”mayappeartohavemultipleusagealthoughtheyareessentiallythesame.Moredetailsaboutthisshouldbelookedforinmoreadvancedtextbooksondifferentialgeometry.

1.4.9 ChristoffelSymbolsforCurvesandSurfaces

TheChristoffelsymbolsofthefirstkindforageneralnDspace(n > 1)aredefinedby:(62)[ij, l] = (∂jgil + ∂igjl − ∂lgij) ⁄ 2wheretheindexedgisthecovariantformofthemetrictensorofthegivenspace.WeremarkthattheLatinindicesusedinthepreviousequation,aswellasthenextfourequations,donotimplya3DspaceandhencetheequationsandthemetricarenotspecifictosuchaspaceastheyapplytoanynDspace(n ≥ 2).TheChristoffelsymbolsofthesecondkind,whichmayalsobecalledaffineconnections,areobtainedbyraisingthethirdindexoftheChristoffelsymbolsofthefirstkind,thatis:(63)Γkij = gkl[ij, l] = (gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij)wheretheindexedgisthemetrictensorofthegivenspaceinitscontravariantandcovariantformswithimpliedsummationoverl.Similarly,theChristoffelsymbolsofthefirstkindcanbeobtainedfromtheChristoffelsymbolsofthesecondkindbyreversingtheaboveprocessthroughloweringtheupperindex,thatis:(64)gkmΓkij = gkmgkl[ij, l] = δlm[ij, l] = [ij, m]whereδlmisthemixedformoftheKroneckerdeltatensorforthespace.ItisnoteworthythattheChristoffelsymbolsofthefirstandsecondkindaresymmetricintheirpairedindices,thatis:(65)[ij, k] = [ji, k](66)Γkij = ΓkjiFor1Dspaces,theChristoffelsymbolsarenotdefined.TheChristoffelsymbolsofthefirstkindfora2Dsurfacearegivenby:(67)[11, 1] = (∂ua11) ⁄ 2 = Eu ⁄ 2(68)[11, 2] = ∂ua12 − (∂va11) ⁄ 2 = Fu − (Ev ⁄ 2)(69)[12, 1] = (∂va11) ⁄ 2 = Ev ⁄ 2 = [21, 1](70)[12, 2] = (∂ua22) ⁄ 2 = Gu ⁄ 2 = [21, 2](71)[22, 1] = ∂va12 − (∂ua22) ⁄ 2 = Fv − (Gu ⁄ 2)(72)[22, 2] = (∂va22) ⁄ 2 = Gv ⁄ 2wheretheindexedaaretheelementsofthesurfacecovariantmetrictensor(referto3.3↓)andE, F, Garethecoefficientsofthefirstfundamentalform(referto3.5↓).Thesubscriptsuandvwhichsuffixthecoefficientsstandforpartialderivativesofthesecoefficientswithrespecttothesevariables(i.e.∂ ⁄ ∂uand∂ ⁄ ∂v).Asindicatedbeforeandwillbeseeninthefuture,E = a11,F = a12 = a21andG = a22,andhencetheequalitiesinthepreviousequationsarejustified.TheaboverelationscanbeobtainedfromthedefinitionoftheChristoffelsymbolsofthefirstkind(Eq.62↑)usingthecoefficientsofthemetrictensorandfirstfundamentalformofthesurface.Inorthogonalcoordinatesystems,F = a12 = a21 = 0identicallyandhencetheaboveformulaewillbesimplifiedaccordinglybydroppinganyterminvolvingthederivativesofthesecoefficients.Wenotethat“orthogonalcoordinatesystem”hereandinsimilarcontextsmeansasystemwhosecoordinatecurvesareorthogonaleverywhere.Therefore,theaboveconditionisfullyjustifiedsinceE1andE2basisvectors,whicharethetangentstotheuandvcoordinatecurves,willbeorthogonalinsuchasystemandhencethedotproduct,seeninEq.235↓,willvanishaccordingly.TheChristoffelsymbolsofthesecondkindfora2Dsurfacearegivenby:(73)Γ111 = (a22∂ua11 − 2a12∂ua12 + a12∂va11) ⁄ (2a) = (GEu − 2FFu + FEv) ⁄ (2a)(74)Γ211 = (2a11∂ua12 − a11∂va11 − a12∂ua11) ⁄ (2a) = (2EFu − EEv − FEu) ⁄ (2a)(75)Γ112 = (a22∂va11 − a12∂ua22) ⁄ (2a) = (GEv − FGu) ⁄ (2a) = Γ121(76)Γ212 = (a11∂ua22 − a12∂va11) ⁄ (2a) = (EGu − FEv) ⁄ (2a) = Γ221(77)Γ122 = (2a22∂va12 − a22∂ua22 − a12∂va22) ⁄ (2a) = (2GFv − GGu − FGv) ⁄ (2a)(78)Γ222 = (a11∂va22 − 2a12∂va12 + a12∂ua22) ⁄ (2a) = (EGv − 2FFv + FGu) ⁄ (2a)wherea( = a11a22 − a12a21 = EG − F2)isthedeterminantofthesurfacecovariantmetrictensor,andtheothersymbolsareasexplainedabove.TheserelationscanbeobtainedfromthedefinitionoftheChristoffelsymbolsofthesecondkind(Eq.63↑)usingthecoefficientsofthemetrictensorandfirstfundamentalformofthesurface.Theaboveformulaewillalsobesimplifiedinorthogonalcoordinatesystems,whereF = a12 = a21 = 0identically,bydroppingthevanishingtermsthatinvolvethesecoefficientsortheirderivatives.TheChristoffelsymbolsofthesecondkindfora2Dsurfacemayalsobegivenby:(79)Γ111 =  − [( E2 × ∂1 E1)⋅ n] ⁄ √(a)(80)Γ211 =  + [( E1 × ∂1 E1)⋅ n] ⁄ √(a)(81)Γ112 =  − [( E2 × ∂2 E1)⋅ n] ⁄ √(a) = Γ121(82)Γ212 =  + [( E1 × ∂2 E1)⋅ n] ⁄ √(a) = Γ221(83)Γ122 =  − [( E2 × ∂2 E2)⋅ n] ⁄ √(a)(84)Γ222 =  + [( E1 × ∂2 E2)⋅ n] ⁄ √(a)wheretheindexedEarethesurfacecovariantbasisvectors(see§1.4.5↑and3.2↓),nistheunitnormalvectortothesurfaceandaisthedeterminantofthesurfacecovariantmetrictensorasdefinedabove.Itisworthnotingthat:√(a) = (E1 ×  E2)⋅ n,aswewillseelaterinthebook.SincetheChristoffelsymbolsofbothkindsaredependentonthemetriconly,ascanbeseenfromtheaboveequations(Eqs.67↑-72↑andEqs.73↑-78↑)aswellasfromthedefinitionsofthesesymbols(seeEqs.62↑and63↑),theyrepresentintrinsicpropertiesofthesurfacegeometryandhencetheyarepartofitsintrinsicgeometry.TheinvolvementofextrinsicparametersinthedefinitionsgivenbyEqs.79↑-84↑doesnotaffecttheirintrinsicqualification,asexplainedearlier.TheChristoffelsymbolsofthefirstkindarelinkedtothesurfacecovariantbasisvectorsbythefollowingrelation:(85)[αβ, γ] = (∂Eα ⁄ ∂uβ)⋅Eγwhereα, β, γ = 1, 2.Thisrelationmayalsobewrittenas:(86)[αβ, γ] = rαβ⋅rγwhereα, β, γ = 1, 2andthesubscriptsrepresentpartialderivativeswithrespecttothevariablesrepresentedbythesecoordinateindices.Thelastequationmayprovideaneasierformtoremembertheseformulae.OnapplyingtheindexraisingoperatortoEq.85↑,weobtainasimilarexpressionfortheChristoffelsymbolsofthesecondkind,thatis:(87)Γγαβ = (∂Eα ⁄ ∂uβ)⋅Eγwhereα, β, γ = 1, 2andEγisthecontravariantformofthesurfacebasisvectors(see§3.2↓).

1.4.10 Riemann-ChristoffelCurvatureTensorTheRiemann-Christoffelcurvaturetensorisanabsoluterank-4tensorthatcharacterizesimportantpropertiesofspaces,including2Dsurfaces,andhenceitplaysanimportantroleindifferentialgeometry.Thetensorisused,forinstance,totestforthespaceflatness(see§1.4.6↑).TherearetwokindsofRiemann-Christoffelcurvaturetensor:firstandsecond.TheRiemann-Christoffelcurvaturetensorofthefirstkindisatype(0, 4)tensorwhiletheRiemann-Christoffelcurvaturetensorofthesecondkindisatype(1, 3)tensor.Shiftingfromonekindtotheotherisachievedbyusingtheindexshiftingoperator.ThefirstandsecondkindsoftheRiemann-Christoffelcurvaturetensoraregivenrespectivelyby:(88)Rijkl = ∂k[jl, i] − ∂l[jk, i] + [il, r]Γrjk − [ik, r]Γrjl(89)Rijkl = ∂kΓijl − ∂lΓijk + ΓrjlΓirk − ΓrjkΓirlwheretheindexedsquarebracketsandΓaretheChristoffelsymbolsofthefirstandsecondkindofthegivenspace,asdefinedin§1.4.9↑.Weremarkthatintheabovetwoequations,aswellasinthefollowingequationsinthepresentandthenextsubsection,theLatinindicesdonotnecessarilyrangeover1, 2, 3astheseequationsarevalidforgeneralnDmanifolds(n ≥ 2)includingsurfaces(n = 2)andspacesofhigherdimensionality(n > 3).AnotherremarkisthattheRiemann-Christoffelcurvaturetensorofthefirstkindisanti-symmetricinitsfirsttwoindicesandinitslasttwoindicesandblocksymmetricwithrespecttothesetwosetsofindices,thatis:(90)Rijkl =  − RjiklRijkl =  − RijlkRijkl =  + RklijFromEqs.88↑and89↑,itcanbeseenthattheRiemann-ChristoffelcurvaturetensordependsexclusivelyontheChristoffelsymbolsofthefirstandsecondkindwhicharebothdependentonthemetrictensor(orthefirstfundamentalform,see§3.5↓)anditspartialderivativesonly.Hence,the

Riemann-Christoffelcurvature,asrepresentedbytheRiemann-Christoffelcurvaturetensor,isanintrinsicpropertyofthemanifold.SincetheRiemann-Christoffelcurvaturetensordependsonthemetricwhich,ingeneralcurvilinearcoordinates,isafunctionofposition,theRiemann-Christoffelcurvaturetensorfollowsthisdependencyonposition.TheRiemann-Christoffelcurvaturetensorvanishesidenticallyiffthespaceisgloballyflatfromintrinsicview;otherwisethespaceiscurved(see§1.4.6↑).Hence,theRiemann-Christoffelcurvaturetensorvanishesidenticallyover1Dmanifoldsasrepresentedbysurfaceandspacecurves.Aswewillseein§2.3↓,surfaceandspacecurvesareintrinsicallyEuclidean.Similarly,theRiemann-Christoffelcurvaturetensorvanishesidenticallyoverplanesurfaces.Moregenerally,asurfaceisisometrictotheEuclideanplaneifftheRiemann-Christoffelcurvaturetensoriszeroateachpointonthesurfacesinceitisintrinsicallyflat.ThelaststatementalsoappliesiftheGaussiancurvatureofthesurfacevanishesidenticallyduetothelinkbetweentheRiemann-ChristoffelcurvaturetensorandtheGaussiancurvature(referto§4.5↓andseeEq.92↓).The2DRiemann-Christoffelcurvaturetensorhasonlyonedegreeoffreedomandhenceitpossessesasingleindependentnon-vanishingcomponentwhichisrepresentedbyR1212.Hence,fora2DRiemannianspacewehave:(91)R1212 = R2121 =  − R1221 =  − R2112whilealltheothercomponentsofthetensorareidenticallyzero.ThesignsintheequalitiesofEq.91↑arejustifiedbytheaforementionedanti-symmetricrelationsoftheRiemann-Christoffeltensorinitsindices(seeEq.90↑).Thevanishingoftheothercomponents(e.g.R1111andR1121)canalsobeexplainedbytheanti-symmetricrelationssincethesecomponentscontaintwoidenticalanti-symmetricindices.Basedontheabovefacts,theRiemann-Christoffelcurvaturetensorcanbeexpressedintensornotationby:(92)Rαβγδ = R1212ϵαβϵγδ = (R1212 ⁄ a)εαβεγδ = Kεαβεγδwheretheindexedϵaretherelativepermutationtensors,theindexedεaretheabsolutepermutationtensors,andKistheGaussiancurvature(see§4.5↓)whoseexpressionisobtainedfromEq.356↓.Thenon-vanishingcomponentofthe2DRiemann-Christoffelcurvaturetensor,R1212,maybegiveninexpandedformby:(93)R1212 = (1 ⁄ 2)(2∂12a12 − ∂22a11 − ∂11a22) + aαβ(Γα12Γβ12 − Γα11Γβ22)where∂αβ ≡ ∂2 ⁄ (∂uα∂uβ)withα, β = 1, 2,theindexedaarethecoefficientsofthesurfacecovariantmetrictensor,andtheChristoffelsymbolsarebasedonthesurfacemetric.

1.4.11 RicciCurvatureTensorandScalarTheRiccicurvaturetensorofthefirstkind,whichisanabsoluterank-2covariantsymmetrictensor,isobtainedbycontractingthecontravariantindexwiththelastcovariantindexoftheRiemann-Christoffelcurvaturetensorofthesecondkind,thatis:(94)Rij = Raija = ∂jΓaia − ∂aΓaij + ΓabjΓbia − ΓabaΓbijTheRiccitensorofthesecondkindisobtainedbyraisingthefirstindexoftheRiccitensorofthefirstkindusingtheindexraisingoperator.For2Dspaces,theRiccicurvaturetensorofthefirstkindisrelatedtotheRiemann-Christoffelcurvaturetensorbythefollowingrelations:(95)(R11 ⁄ a11) = (R12 ⁄ a12) = (R21 ⁄ a21) = (R22 ⁄ a22) =  − (R1212 ⁄ a)wheretheindexedaaretheelementsofthe2Dcovariantmetrictensor(see§3.3↓)andthebareaisitsdeterminant.Asthe2DRiemann-Christoffelcurvaturetensorhasonlyoneindependentnon-vanishingcomponent,thelastequationprovidesafulllinkbetweentheRiccicurvaturetensorandtheRiemann-Christoffelcurvaturetensor.SinceK = (R1212 ⁄ a)(seeEq.356↓),theaboverelationsalsolinktheRiccitensortotheGaussiancurvature.TheRicciscalarℛ,whichisalsocalledthecurvaturescalarandthecurvatureinvariant,istheresultofcontractingtheindicesoftheRiccicurvaturetensorofthesecondkind,thatis:(96)ℛ = RiiHence,itisarank-0tensor.Asseenfromtheaboverelations,theRiccicurvaturetensorandcurvaturescalararepartoftheintrinsicgeometryofthemanifold.

1.5 ExercisesExercise1.Giveabriefdefinitionofdifferentialgeometryindicatingtheotherdisciplinesofmathematicstowhichdifferentialgeometryisintimatelylinked.Exercise2.Asurfaceembeddedina3Dspacecanberegardedasa2Dandasa3Dobjectatthesametime.Discussthisbriefly.Fromthesameperspective,discussalsothestateofacurveembeddedinasurfacewhichinitsturnisembeddedina3Dspace.Exercise3.Whatarethefollowingsymbols:[αβ, γ],[ij, k],Γγ

αβandΓkij?

WhatisthedifferencebetweenthosewithGreekindicesandthosewithLatinindices?Exercise4.Whatistherelationbetweenthecoefficientsofthesurfacecovariantmetrictensorandthesurfacecovariantcurvaturetensorononehandandthecoefficientsofthefirstandsecondfundamentalformsontheother?Whatarethesymbolsrepresentingallthesecoefficients?Exercise5.Whatisthedifferencebetweenthelocalandglobalpropertiesofamanifold?Giveanexampleforeach.Whatarethecolloquialtermsusedtolabelthesetwocategories?Exercise6.Whatisthemeaningof“intrinsic”and“extrinsic”propertiesofamanifold?Giveanexampleforeach.Exercise7.Explaintheconceptof“2Dinhabitant”andhowitisusedtoclassifythepropertiesofaspacesurface.Exercise8.Findtheequationofaplanepassingthroughthepoints:(1, 2, 0),(0,  − 3, 1.5)and(1, 0,  − 1).Whatisthenormalunitvectortothisplane?Exercise9.Isthenormalunitvectorofaplanesurfaceunique?Exercise10.Definebrieflyeachoneofthefollowingterms:surfaceofrevolution,meridiansandparallels.Exercise11.Provethatthemeridiansandparallelsofasurfaceofrevolutionaremutuallyperpendicularattheirpointsofintersection.Exercise12.Statetheparametricequationsofthefollowinggeometricshapes:torus,hyperboloidofonesheet,andhyperbolicparaboloid.Exercise13.Writedowntheparametricequationsofacircleintheuvplanecenteredatpoint(a, b)withradiusr = cwherea, b, carerealconstantsandc > 0.Exercise14.FindtheparametricequationsofanellipseinthexyplanecenteredattheoriginofcoordinateswithA = 5andB = 3whereA, Barethesemi-majorandsemi-minoraxes.Exercise15.Asurfaceofrevolutionmayberepresentedlocallyina3Dspacebythefollowingform:r(u, v) = (ucosv, usinv, f(u, v))wherefisacontinuousfunction.Determinetheequationsrepresentingtheparallelsandmeridiansofthissurface.Exercise16.Findtheparametricequationsofacurveformedbytheintersectionofthesurfacesrepresentedby:r1(u, v) = (u, u2, v)andr2(u, v) = (u, v, u2)where − ∞ < u, v <  + ∞.Exercise17.Writedownthegeneralformoftheparametricequationsofeachofthefollowingsurfaces:hyperboloidoftwosheets,paraboliccylinder,catenoid,monkeysaddleandpseudo-sphere.Exercise18.Sketchthefollowing(usinga3Dcomputergraphicpackageifavailable):(a)astraightlinepassingthroughthepoint(11,  − 5, 6.3)andparalleltothevector( − 3,  − 1.8, 6.5)(b)aplanepassingthroughthepoint(6,  − 8.2,  − 7)withanormalvector(3,  − 1.6,  − 2.5).Exercise19.Asurfaceisparameterizedby:r(u, v) = (asinhucosv, bsinhusinv, ccoshu).Whatisthenameofthissurface?Whatistheconditionforthissurfacetobeasurfaceofrevolutionaroundthethirdspatialaxis?Exercise20.Findtheequationofthestraightlinepassingthroughthepoint( − 6, 3.1, 8.4)andthepoint(1, 0,  − 3).Exercise21.Classifythefollowingascurvesorsurfaces:ellipsoid,ellipticparaboloid,catenary,helicoid,enneperandtractrix.Exercise22.Makeasimplesketchforeachoneofthegeometricshapesinthepreviousquestion.Useacomputergraphicpackageifconvenient.

Exercise23.Provethatv × (dv ⁄ dt) = 0iffthedirectionofthevectorv(t)isconstant.Exercise24.Define“Eulercharacteristic”statingtheequationthatlinksittothenumberofvertices,facesandedgesofapolyhedron.Exercise25.ExplainhowtheEulercharacteristicisdefinedfornon-polyhedralcompactorientablesurfacessuchasellipsoids.Exercise26.Whatisthetopologicalmeaningof“genusofasurface”?Exercise27.Giveexamplesforsurfacesofgenus0,1,2and3fromcommongeometricshapesotherthanthosegiveninthetext.Exercise28.ByusingtheEulerformula,calculatetheEulercharacteristicχofthefollowingsurfaces:(a)parallelepiped(b)dodecahedron(c)icosahedron.Showyourworkindetail.Exercise29.Byusingpolygonaldecomposition,calculatetheEulercharacteristicχofthefollowingsurfaces:(a)sphere(b)ellipsoid(c)torus.Showyourworkindetailwithsimplesketchestodemonstratethepolygonaldecompositionineachcase.Exercise30.Whatisthegenusofthesurfacesinthepreviousquestion?Exercise31.WhatistheCartesianformoftheequationofaspherecenteredatpoint(a, b, c)withradiusr = dwherea, b, c, darerealconstantsandd > 0?Exercise32.Explainbrieflythemeaningofthefollowingterms:bicontinuousfunction,surfaceofclassCn,andsufficientlysmoothcurve.Exercise33.Explainindetail,usingequationsandsimplesketches,theconceptof“deletedneighborhood”in1Dand2Dflatandcurvedspacesasseenfromtheambientspace.Exercise34.Howcanweextendtheconceptof“deletedneighborhood”tospacesofdimensionalityhigherthan2?Exercise35.Findtheequationofaconegeneratedbyrotatingthelinez =  − 2xaroundthez-axis.Exercise36.Derivetheparametricequationsofahelixrotatingaroundthez-axis,passingthroughthepoint(3, 0, 0)andclimbing(ordescending)5.3unitsinthez-directionasitmakesa4πturnaroundthez-axis.Exercise37.Define“positivedefinite”inwordsandbystatingthemathematicalconditionsforaquadraticexpressiontobepositivedefinite.Exercise38.DescribeorthogonalcoordinatetransformationsandhowtheyarecharacterizedbytheirJacobian.Exercise39.Statethedifferencebetweenpositiveandnegativeorthogonaltransformations.Exercise40.Findasetofparametricequationsrepresentingacylindergeneratedbyrotatingastraightlineparalleltothez-axisandpassingthroughthepoint(2.5,0,0)aroundthez-axis.Exercise41.Whatistheunitnormalvectortothesurfaceofasphere,centeredontheoriginofcoordinateswithradiusr,atapointonitssurfacewithcoordinates(xP, yP, zP)?Considerthepossibilityofhavingmorethanonenormalvectoratthatpoint.Exercise42.What“coordinatecurves”means?Whataretheothernamesgiventothesecurves?Exercise43.DefineregularrepresentationofaclassCnsurfacepatchina3DEuclideanspacestatingitsmathematicalconditions.Exercise44.Usingaparametricrepresentationoftheellipticparaboloid,showthatitisaregularsurface.Exercise45.Whatarethereasonsforhavingasingularpointonaspacesurface?Exercise46.Aspherecenteredattheoriginofcoordinatescanberepresentedparametricallyby:r(θ, φ) = a(sinθcosφ, sinθsinφ, cosθ)wherea > 0isaconstant,0 ≤ θ ≤ πand0 ≤ φ < 2π.Atwhatpoints,ifany,thisrepresentationisnotregular?Exercise47.Howamathematicalcorrespondencecanbeestablishedbetweenpointsontwodifferentcurvesandtwodifferentsurfaces?Exercise48.Statethemathematicalconditionswhicharesatisfiedbytheintrinsicdistancebetweentwopointsonasmoothconnectedsurface.

Exercise49.Isitguaranteedthatanarcofminimumlengthbetweentwospecificpointsonasurfacedoesexistanditisunique?Exercise50.Provethethreepropertiesofintrinsicdistance,i.e.symmetry,triangleinequality,andpositivedefiniteness.Exercise51.Showthattheintrinsicdistancedbetweentwopointsisinvariantunderalocalisometricmappingf,i.e.d(f(P1), f(P2)) = d(P1, P2).Exercise52.Findtheintrinsicdistanceontheunitspherecenteredattheoriginofcoordinatesbetweenthepoint(1, 0, 0)andthepoint(1 ⁄ √(3), 1 ⁄ √(3), 1 ⁄ √(3)).Exercise53.Whatistheintrinsicdistancebetweenthetwopointsofthelastquestionina3DEuclideanspacethatenclosesthesphere?Exercise54.Definethebasisvectorsandstatetheirroles.Exercise55.Describeindetailthetwomainsetsofspacebasisvectorsindifferentialgeometryrelatedtocurvesandsurfaces.Arethereanyothersetsofbasisvectors?Exercise56.Arethebasisvectorsnecessarilyofunitlengthand/ormutuallyorthogonal?Ifnot,giveexamplesofbasisvectorswhicharenotofunitlengthand/ormutuallyorthogonal.Exercise57.Define,inmathematicalterms,flatandcurvedspacesgivingexamplesforeach.Exercise58.StateasufficientandnecessaryconditionforannDspacetobeflat.Exercise59.IsitnecessarythatannDcurvedspacepossessesuniversallyconstantcurvature?Ifnot,giveanexampleofaspacewithvariablecurvatureinsignandmagnitude.Exercise60.Whatisthelocusofthepoints(ifany)whicharesharedbetweenthexyplaneandthefollowingsurfaces:(a)aspherecenteredat(0, 0, 5)withradiusr = 6(b)aspherecenteredat(1, 1, 1)withradiusr = 1.5(c)aplanepassingthroughthepoint(5,  − 9.6, 0)withaunitnormalvector(0, 0,  − 1)?Exercise61.DescribethecommonlyusedapproachforinvestigatingtheRiemanniangeometryofcurvedmanifolds.Exercise62.Giveabriefdefinitionofhomogeneouscoordinatesystemsgivingacommonexampleofsuchsystems.Exercise63.WhatistherelationbetweentheChristoffelsymbolsofthefirstkindandtheChristoffelsymbolsofthesecondkind?Exercise64.Writethemathematicalexpressionsforthesymbols[12, 1]andΓ1

22ofasurfaceintermsofthecoefficientsofthesurfacemetrictensor.Exercise65.UsingthedefinitionoftheChristoffelsymbolsofthefirstkindandtherulesoftensors,deriveEqs.68↑and71↑.Exercise66.UsingthedefinitionoftheChristoffelsymbolsofthesecondkindandtherulesoftensors,deriveEqs.76↑and78↑.Exercise67.StatethemathematicalrelationscorrelatingtheChristoffelsymbolsofthefirstandsecondkindtothesurfacebasisvectorsandtheirderivatives.Exercise68.WhatistherelationbetweentheRiemann-ChristoffelcurvaturetensorandtheGaussiancurvatureofasurface?Exercise69.Howmanyindependentnon-vanishingcomponentsthe2DRiemann-Christoffelcurvaturetensorpossesses?Exercise70.WhatisthesignificanceofhavinganidenticallyvanishingRiemann-Christoffelcurvaturetensorona2Dsurface?Exercise71.ShowthattheRiemann-Christoffelcurvaturetensorisanti-symmetricinitsfirsttwoindicesandinitslasttwoindicesandblocksymmetricwithrespecttothesetwosetsofindices.Exercise72.ProveEqs.85↑and87↑.Exercise73.WhatistherankoftheRiccicurvaturetensor?Exercise74.StatethemathematicalrelationthatlinkstheRiccicurvaturetensorofthefirstkindtotheChristoffelsymbolsofthesecondkindandtheirpartialderivatives.Exercise75.WhatistherelationbetweentheelementsoftheRiccicurvaturetensorofthefirstkindandtheGaussiancurvature?Exercise76.HowdoyouobtaintheRicciscalarfromtheRiemann-Christoffelcurvaturetensorofthefirstkind?Explainyouranswerstepbystep.

Chapter2CurvesinSpaceInthischapter,weinvestigatecurvesresidinginahigherdimensionalityspaceandhowtheyarecharacterized.Weshouldfirstremarkthat“space”inthistitleisgeneralandhenceitincludessurfacesinceitisa2Dspace,asexplainedin§1.2↑.

2.1 GeneralBackgroundaboutCurvesInsimpleterms,aspacecurveisasetofconnectedpoints(seeFootnote7in§8↓)intheembeddingspacesuchthatanytotallyconnectedsubsetofitcanbetwistedintoastraightlinesegmentwithoutaffectingtheneighborhoodofanypoint.Moretechnically,acurveisdefinedasadifferentiableparameterizedmappingbetweenanintervalofthereallineandaconnectedsubsetoftheembeddingspace,thatisC(t):I→ℝnwhereCrepresentsaspacecurvedefinedontheintervalI ⊆ ℝandparameterizedbythevariablet ∈ I.Hence,differentparameterizationsofthesame“geometriccurve”willleadtodifferent“mappingcurves”.Theimageofthemappingintheembeddingspaceisknownasthetraceofthecurve;hencedifferentmappingcurvescansharethesametrace.ThecurvemayalsobedefinedasatopologicalimageofarealintervalandmaybelinkedtotheconceptofJordanarc.WenotethatJordanarcsorJordancurvesmaybedefinedasinjectivemappingswithnoselfintersection.Spacecurvescanbedefinedsymbolicallyindifferentways;themostcommonoftheseisparametricallywherethethreespacecoordinatesofthecurvepointsaregivenasfunctionsofarealvaluedparameter,e.g.xi = xi(t)wheret ∈ ℝisthecurveparameterandi = 1, 2, 3.Theparametertmayrepresenttimeorarclengthorevenanarbitrarilydefinedquantity.Similarly,surfacecurvesaredefinedparametricallywherethetwosurfacecoordinatesaregivenasfunctionsofarealvaluedparameter,e.g.uα = uα(t)withα = 1, 2.Thesurfacecoordinatescanthenbemapped,throughanothermappingrelation,ontothespatialrepresentationofthesurfaceintheenvelopingspace.Parameterizedcurvesareorientedobjectsastheycanbetraversedinonedirectionortheotherdependingonthesenseofincreaseoftheirparameter.Therearetwomaintypesofcurveparameterization:parameterizationbyarclengthandparameterizationbysomethingelsesuchastime.TheconditionforaspacecurveC(t):I→ℝ3,wheret ∈ IisthecurveparameterandI ⊆ ℝisanintervaloverwhichthecurveisdefined,tobeparameterizedbyarclengthisthat:foralltwehave|dr ⁄ dt| = 1wherer(t)isthepositionvectorrepresentingthecurveintheambientspace.Asaconsequenceofthis,parameterizationbyarclengthisequivalenttotraversingthecurvewithaconstantunityspeed.Hence,usingaparameterizationbysomethingotherthanarclengthmaybeconsideredastraversingthecurvewithvaryingornon-unityspeed.Theadvantageofparameterizationbyarclengthisthatitconfinestheattentiononthegeometryofthecurveratherthanotherfactors,whichareusuallyirrelevanttothegeometricinvestigation,suchasthetemporalrateoftraversingthecurve.Furthermore,itusuallyresultsinamoresimplemathematicalformulation,aswewillseelaterinthebook.Theparametersymbolwhichisusednormallyforparameterizationbyarclengthiss,whiletisusedtorepresentageneralparameterwhichcouldbearclengthorsomethingelse.Thisnotationisfollowedinthepresentbook.Forcurvesparameterizedbyarclength,thelengthLofasegmentbetweentwopointsonthecurvecorrespondingtos1ands2isgivenbythesimpleformula:(97)

Parameterizationbyarclengthsmaybecallednaturalparameterizationofthecurveandhencesiscallednaturalparameter.Naturalparameterizationisnotunique;howeveranyothernaturalparameteršisrelatedtoagivennaturalparametersbythefollowingrelation:(98)š = ±s + cwherecisarealconstantandhencetheabove-statedcondition|dr ⁄ dt| = 1remainsvalid(seeFootnote8in§8↓).Thismaybestatedinadifferentwaybysayingthatnaturalparameterizationwitharclengthsisuniqueapartfromthepossibilityofhavingadifferentsenseoforientationandanadditiveconstanttos.Naturalparameterizationmayalsobeusedforparameterizationbyaparameterwhichisproportionaltosandhencethetransformationrelationbetweentwonaturalparametersbecomes:(99)š = ±ms + cwheremisanotherrealconstant.Thetwoparameterizationsthendiffer,apartfromthesenseoforientationandtheconstantshift,bythelengthscalewhichcanbechosenarbitrarily.Consequently,naturalparameterizationwillbeequivalenttotraversingthecurvewithaconstantspeednotnecessarilyofunitymagnitude.Thismaybebasedonthevisionofextendingtheaforementionedbenefitsofnaturalparameterizationbyscaling,i.e.bychoosingadifferentlengthscaleanaturalparameterizationwillbeobtainedspontaneously.Inthisbook,naturalparameterizationisrestrictedtoparameterizationwitharclengths.InageneralnDspace,thetangentvectortoaspacecurve,representedparametricallybythespatialrepresentationr(t)wheretisageneralparameter,isgivenby(dr ⁄ dt).AvectortangenttoaspacecurveatPisanon-trivialscalarmultipleof(dr ⁄ dt)andhenceitcandifferinmagnitudeanddirectionfrom(dr ⁄ dt)asitcanbeparalleloranti-parallelto(dr ⁄ dt).Thetangentvectortoasurfacecurve,representedparametricallyby:C(u(t), v(t))whereuandvarethesurfacecoordinatesandtisageneralparameter,isgivenby:(100)dr ⁄ dt = (∂r ⁄ ∂u)(du ⁄ dt) + (∂r ⁄ ∂v)(dv ⁄ dt)wherer(u(t), v(t))isthespatialrepresentationofCandwhereallthesequantitiesaredefinedandevaluatedataparticularpointonthecurve.Thelastequationcanbecastcompactly,usingtensornotation,as:(101)dxi ⁄ dt = (∂xi ⁄ ∂uα)(duα ⁄ dt) = xiα(duα ⁄ dt)wherei = 1, 2, 3,α = 1, 2and(u1, u2) ≡ (u, v).AspacecurveC(t):I → ℝ3,whereI ⊆ Randt ∈ Iisageneralparameter,is“regularatpointt0”iffr (t0)existsandr (t0) ≠  0wherer(t)isthespatialrepresentationofCandtheoverdotstandsfordifferentiationwithrespecttothegeneralparametert.Thecurveis“regular”iffitisregularateachinteriorpointinI.Onaregularparameterizedcurvethereisaneighborhoodtoeachpointinitsdomaininwhichthecurveisinjective.OntransformingasurfaceSbyadifferentiableregularmappingfofclassCntoasurfaceS,aregularcurveCofclassCnonSwillbemappedonaregularcurveCofclassCnonSbythesamefunctionalmappingrelation,thatisr(t) = f(r(t))wherethebarredandunbarredr(t)arethespatialparametricrepresentationsofthetwocurvesonthebarredandunbarredsurfaces.ThetangentlinetoasufficientlysmoothcurveatoneofitsregularpointsPisastraightlinepassingthroughPbutnotthroughanypointinadeletedneighborhoodofP.Moretechnically,thetangentlinetoacurveCatagivenregularpointPisastraightlinepassingthroughPandhavingthesameorientationasthetangentvector(dr ⁄ dt)ofCatP.ThetangentlinetoacurveatagivenpointPonthecurvemayalsobedefinedgeometricallyasthelimitofasecantlinepassingthroughPandanotherneighboringpointonthecurveastheotherpointconverges,whilestayingonthecurve,tothetangentpoint.Thesedifferentdefinitionsareequivalentastheyrepresentthesameentity.Itisnoteworthythatthetangentlineofasmoothcurve,wheresuchatangentdoesexit,isunique.Wealsonotethatthegeometricdefinitionmaybeusefulinsomecaseswheretheanalyticaldefinitiondoesnotapply.Anon-trivialvectorvissaidtobetangenttoaregularsurfaceS(see§1.4.3↑)atagivenpointPonSifthereisaregularcurveConSpassingthroughPsuchthatv = dr ⁄ dtwherer(t)isthespatialrepresentationofCand(dr ⁄ dt)isevaluatedatP(alsosee3.1↓forfurtherdetails).Infact,anyvectorv = c(dr ⁄ dt)wherec ≠ 0isarealnumber,isatangentalthoughitmaynotbethetangent.AperiodiccurveCisacurvethatcanberepresentedparametricallybyacontinuousfunctionoftheformr(t + T) = r(t)whereristhespatialrepresentationofC,tisarealgeneralparameterandTisarealconstantcalledthefunctionperiod.Circlesandellipses(Fig.22↓)areprominentexamplesofperiodiccurveswheretheycanberepresentedparametricallyby:(102)r(t) = (acost, asint)(circle)(103)r(t) = (acost, bsint)(ellipse)whereaandbarerealpositiveconstantsandt ∈ ℝ.Duetotheperiodicityofthetrigonometricfunctions,theseequationssatisfythecondition:r(t + 2π) = r(t).Hence,circlesandellipsesareperiodiccurveswithaperiodof2π.

Figure22 Circle(leftframe)andellipse(rightframe)andtheirmainparameters.Aclosedcurveisacontinuousperiodiccurvedefinedoveraminimumofoneperiod.Wenotethatperiodicityisnotanecessaryrequirementforthedefinitionofclosedcurvesasthecurvescanbedefinedoverasingleperiodwithoutbeingconsideredassuchorbyfunctionsofnon-periodicnature.Closedcurvesmayberegardedastopologicalimagesofcircles.Acurveisdescribedasaplanecurveifitcanbeembeddedentirelyinaplanewithnodistortion.Orthogonaltrajectoriesofagivenfamilyofcurvesisafamilyofcurvesthatintersectthegivenfamilyperpendicularlyattheirintersectionpoints.Anycurvecanbemappedisometricallytoastraightlinesegmentwherebotharenaturallyparameterizedbyarclength.Fromthelaststatementplusthefactthatisometrictransformationisanequivalencerelation(see§6.5↓),itcanbeconcludedthatanytwospaceandsurfacecurvescanbeconnectedbyanisometricrelation.

2.2 MathematicalDescriptionofCurvesLethaveaspacecurveofclassC2ina3DRiemannianmanifoldwithagivenmetricgij(i, j = 1, 2, 3).Thecurveisparameterizednaturallybysrepresentingarclength.Asstatedearlier,wechoosetoparameterizethecurvebystohavesimplerformulaealthough,forthesakeofcompletenessandgenerality,otherformulaebasedonamoregeneralparameterizationwillalsobegiven.Thecurvecanthereforeberepresentedby:(104)xi = xi(s)wherei = 1, 2, 3andtheindexedxrepresentthespacecoordinates.Thisisequivalentto:(105)r(s) = xi(s)EiwhereristhespatialrepresentationofthespacecurveandEiarethespacebasisvectors.Threemutuallyperpendicularvectorseachofunitlengthcanbedefinedateachregularpointoftheabove-describedspacecurve:tangentT,normalNandbinormalB(seeFig.23↓).Aswellascharacterizingthecurve,thesevectorscanserveasamovingcoordinatesystemfortheembeddingspaceasindicatedearlier.Forsimplicity,clarityandpotentiallackoffamiliaritywithtensordifferentiation(see§7↓)atthisstage,thefollowingismostlybasedonassumingaEuclideanspacecoordinatedbyarectangularCartesiansystemalthoughsupplementaryremarksrelatedtomoregeneralspaceandcoordinatesareaddedwhennecessary.Wealsouseamixoftensorandsymbolicnotationsaseachhascertainadvantagesandtofamiliarizethereaderwithbothnotationssincedifferentauthorsusedifferentnotations.

Figure23 ThevectorsT, N, Bandtheirassociatedplanes(osculatingplaneOP,normalplaneNP,andrectifyingplaneRP)ofacurveCembeddedinaEuclideanspacewitharectangularCartesiancoordinatesystem(seealsoFig.24↓).

TheunitvectortangenttothecurveatagivenregularpointPonthecurveisgivenby(seeFootnote9in§8↓):(106)[T]i = Ti = dxi ⁄ dsForat-parameterizedcurve,wheretisnotnecessarilythearclength,thetangentvectorisgivenby:(107)T = ṙ(t) ⁄ |ṙ(t)|wheretheoverdotrepresentsdifferentiationwithrespecttot.TheunitvectornormaltothetangentTi,andhencetothecurve,atthepointPisgivenby:(108)[N]i = Ni = (dTi ⁄ ds) ⁄ |dTi ⁄ ds| = (dTi ⁄ ds) ⁄ κwhereκisascalarcalledthe“curvature”ofthecurveatthepointPandisdefined,accordingtothenormalizationcondition,by:(109)κ = √([dTi ⁄ ds][dTi ⁄ ds])Forat-parameterizedcurve,thenormalunitvectorisgivenby:(110)N = [ṙ(t) × (r(t) × ṙ(t))] ⁄ [|ṙ(t)||r(t) × ṙ(t)|]ThevectorNisalsocalledtheprincipalnormalvector.BasedonEq.108↑,thisvectorisdefinedonlyonpointsofthecurvewherethecurvatureκ ≠ 0.Also,sinceTisaunitvectorthenitsderivativeisorthogonaltoit,sotheabove-statedfactsareconsistent.WenotethatEq.109↑isbasedonanunderlyingCartesiancoordinatesystem.Forgeneralcurvilinearcoordinates,theformulabecomes:(111)κ = √(gij[δTi ⁄ δs][δTj ⁄ δs])wheregijisthespacecovariantmetrictensorandthenotationofabsolutederivativeisinuse(see§7↓).Thebinormalunitvectorisdefinedas:(112)[B]i = Bi = [κTi + (dNi ⁄ ds)] ⁄ τwhichisalinearcombinationoftwovectorsbothofwhichareperpendiculartoNiandhenceitisperpendiculartoNi.Inthelastequation,thenormalizationscalarfactorτisthe“torsion”whosesignischosentomakeT, N, Barighthandedsystemsatisfyingthecondition:(113)εijkTiNjBk = 1whereεijkisthecovariantabsolutepermutationtensorforthe3Dspace.Weshouldalsoimposetheconditionτ ≠ 0onEq.112↑.Forplanecurves,wherethetorsionvanishesidentically(see§2.3.2↓),andatthepointswithτ = 0oftwistedcurves,thebinormalunitvectorBmaybedefinedgeometricallyorasthecrossproductofTandN,i.e.B = T × N.Forat-parameterizedcurve,thebinormalvectorisgivenby:(114)B = [ṙ(t) × r(t)] ⁄ |ṙ(t) × r(t)|InlinewithmakingT, N, Barighthandedsystem,thereisageometricsignificanceforthesignofthetorsionasitaffectstheorientationofthespacecurve.ItshouldberemarkedthatsomeauthorsreversethesigninthedefinitionofτandthisreversalaffectsthesignsintheforthcomingFrenet-Serretformulae(see§2.5↓).Theconventionthatwefollowinthisbookmayhavecertainadvantages.Atanypointonthespacecurve,thetriadT, N, Brepresentamutuallyperpendicularrighthandedsystemfulfillingthecondition:(115)Bi = [T × N]i = εijkTjNkwhereεijkisthecontravariantabsolutepermutationtensorforthe3Dspace.SincethevectorsinthetriadT, N, Baremutuallyperpendicular,theysatisfytheconditions:(116)T⋅N = T⋅B = N⋅B = 0Moreover,becausetheyareunitvectorstheyalsosatisfytheconditions:(117)T⋅T = N⋅N = B⋅B = 1ItshouldberemarkedthatthetriadT, N, BformwhatiscalledtheFrenetframewhichrepresentsasetoforthonormalbasisvectorsfortheembeddingspace.Thisframeservesasabasisforamovingorthogonalcoordinatesystemonthepointsofthecurve.TheFrenetframevariesingeneralasitmovesalongthecurveandhenceitisafunctionofthepositiononthecurve.ThetriadT, N, BmayalsobecalledtheFrenettrihedronorthemovingtrihedronofthecurve.TheFrenetframecansufferfromproblemsorbecomeundefined,e.g.atnon-regularpointswhereTisundefinedoratinflectionpointswheredT ⁄ ds = 0.ThetangentlineofacurveCatagivenpointPonthecurveisastraightlinepassingthroughPandisparalleltothetangentvector,T,ofCatP.TheprincipalnormallineofacurveCatagivenpointPonthecurveisastraightlinepassingthroughPandisparalleltotheprincipalnormalvector,N,ofCatP.ThebinormallineofacurveCatagivenpointPonthecurveisastraightlinepassingthroughPandisparalleltothebinormalvector,B,ofCatP.Asaconsequence,theequationsofthethreelinescanbegivenbythefollowinggenericform:(118)r = rP + kVPwhereristhepositionvectorofanarbitrarypointontheline,rPisthepositionvectorofthepointP,kisarealvariable( − ∞ < k < ∞)andthevectorVPisthevectorcorrespondingtotheparticularline,thatisVP ≡ Tforthetangentline,VP ≡ Nfortheprincipalnormalline,andVP ≡ Bforthebinormalline.AtanypointPonaspacecurvewheretheFrenetframeisdefined,thetriadT, N, BdefinethreemutuallyperpendicularplaneswhereeachoneoftheseplanespassesthroughthepointPandisformedbyalinearcombinationoftwoofthesevectorsinturn.Theseplanesare:the“osculatingplane”whichisthespanofTandN,the“rectifyingplane”whichisthespanofTandB,andthe“normalplane”whichisthespanofNandBandisorthogonaltothecurveatP(Fig.24↓).Asaresult,theequationsofthethreeplanescanbegivenbythefollowinggenericform:(119)(r − rP)⋅VP = 0whereristhepositionvectorofanarbitrarypointontheplane,rPisthepositionvectorofthepointP,andwhereforeachplanethevectorVPistheperpendicularvectortotheplaneatP,thatisVP ≡ Bfortheosculatingplane,VP ≡ Nfortherectifyingplane,andVP ≡ Tforthenormalplane.Followingthestyleofthedefinitionofthetangentlineofacurveasthelimitofthesecantline(see§2.1↑),theosculatingplanemayalsobedefinedasthelimitingpositionofaplanepassingthroughPandtwootherpointsonthecurveasthetwopointsconvergesimultaneouslyalongthecurvetoP.

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Figure24 Frenetplanes(osculatingplaneOP,normalplaneNP,andrectifyingplaneRP)andbasisvectors(T, N, B)atapointonaspacecurveC.Itisnoteworthythatthepositivesenseofaparameterizedcurve,whichcorrespondstothedirectioninwhichtheparameterincreasesandhencedefinestheorientationofthecurve,canbedeterminedintwooppositeways.WhilethesenseofthetangentTandthebinormalBisdependentonthecurveorientationandhencetheyareinoppositedirectionsinthesetwoways,theprincipalnormalNisthesameasitremainsparalleltothenormalplaneinthedirectioninwhichthecurveisturning.ThisisconsistentwiththefactthatthetriadT, N, Bformarighthandedsystem.

2.3 CurvatureandTorsionofSpaceCurvesThecurvatureandtorsionofspacecurvesmayalsobecalledthefirstandsecondcurvaturesrespectively,andhenceatwistedcurvewithnon-vanishingcurvatureandnon-vanishingtorsionisdescribedasdouble-curvaturecurve.Theexpression√([dsT]2 + [dsB]2),wheredsTanddsBarerespectivelythelengthsofthelineelementcomponentsinthetangentandbinormaldirections,maybedescribedasthetotalorthethirdcurvatureofthecurve.TheequationofLancretstatesthat:(120)(dsN)2 = (dsT)2 + (dsB)2wheredsNisthelengthofthelineelementcomponentintheprincipalnormaldirection.Wenotethattheterm“totalcurvature”isalsousedforsurfaces(see§4.4↓and4.8↓)butthemeaningisobviouslydifferent.Accordingtothefundamentaltheoremofspacecurvesindifferentialgeometry,aspacecurveiscompletelydeterminedbyitscurvatureandtorsion.Moretechnically,givenarealintervalI ⊆ ℝandtwodifferentiablerealfunctions:κ(s) > 0andτ(s)wheres∈I,thereisauniquelydefinedparameterizedregularspacecurveC(s):I → ℝ3ofclassC2withκ(s)andτ(s)beingthecurvatureandtorsionofCrespectivelyandsisitsarclength.Hence,anyothercurvemeetingtheseconditionswillbedifferentfromConlybyarigidmotiontransformation(i.e.translationandrotation)whichdeterminesitspositionandorientationinspace.Ontheotherhand,anycurvewiththeabove-describedpropertiespossessesuniquelydefinedκ(s)andτ(s).Asaconsequenceofthelaststatements,thefundamentaltheoremofspacecurvesprovidestheexistenceanduniquenessconditionsforcurves.Wenoteinthiscontextthatinrigidmotiontransformation,whichmayalsobecalledEuclideanmotion,thedistancebetweenanytwopointsontheimageisthesameasthedistancebetweenthecorrespondingpointsontheinverseimage.Hence,rigidmotiontransformationisaformofisometricmapping.Theequations:κ = κ(s)andτ = τ(s),wheresisthearclength,arecalledtheintrinsicornaturalequationsofthecurve.Thecurvatureandtorsionareinvariantsofthespacecurveandhencetheydonotdependinmagnitudeontheemployedcoordinatesystemorthetypeofparameterization.Whilethecurvatureisalwaysnon-negative(κ ≥ 0),asitrepresentsthemagnitudeofavectoraccordingtotheabove-stateddefinition(seeEqs.108↑,109↑and111↑),thetorsioncanbenegativeaswellaszeroorpositive.Itisworthmentioningthatsomeauthorsdefinethecurvaturevector(see§4.1↓)andtheprincipalnormalvectorofspacecurvesinsuchawaythatitispossibleforthecurvaturetobenegative.Thefollowingaresomeexamplesofthecurvatureandtorsionofanumberofcommonly-occurringsimplecurves:1.Straightline:κ = 0andτ = 0.2.CircleofradiusR:κ = (1 ⁄ R)andτ = 0.Hence,theradiusofcurvature(see§2.3.1↓)ofacircleisitsownradius3.Helixparameterizedbyr(t) = (acos(t), asin(t), bt):κ = a ⁄ (a2 + b2)andτ = b ⁄ (a2 + b2).ItisworthnotingthataspacecurveofclassC3withnon-vanishingcurvatureisahelixifftheratioofitstorsiontocurvatureisconstant.Intheabovethreeexamples,thecurvatureandtorsionareconstantalongthewholecurve.However,ingeneralthecurvatureandtorsionofspacecurvesarepositiondependentandhencetheyvaryfrompointtopoint.Followingtheexampleof2Dsurfaces,a1Dinhabitantofaspacecurvecandetectallthepropertiesrelatedtothearclength.Hence,thecurvatureandtorsion,κandτ,ofthecurveareextrinsicpropertiesforsucha1Dinhabitant.ThisfactmaybeexpressedbysayingthatcurvesareintrinsicallyEuclidean,andhencetheirRiemann-Christoffelcurvaturetensorvanishesidenticallyandtheynaturallyadmit1DCartesiansystemsrepresentedbytheirnaturalparameterizationofarclength.Thisshouldbeobviouswhenconsideringthatanycurvecanbemappedisometricallytoastraightlinewherebotharenaturallyparameterizedbyarclength.Anotherdemonstrationoftheirintrinsic1DnatureisrepresentedbytheforthcomingFrenet-Serretformulae(see§2.5↓).Itisnoteworthythatsomeauthorsresembletheroleofκandτincurvetheorytothesurfacecurvaturetensorbαβinsurfacetheory(see§3.4↓)anddescribeκandτasthecurvetheoreticanaloguesofbαβinsurfacetheory.Inanothercontext,κandτmaybecompared(non-respectively!)withthefirstandsecondfundamentalformsofsurfacesintheirrolesindefiningthecurveandsurfaceinthefundamentaltheoremsofthesestructures(comparetheabovewithwhatiscomingin§3.6↓).ThecurvatureandtorsionalsoplayintheFrenet-Serret

formulaeforspacecurvesasimilarroletotheroleplayedbythecoefficientsofthefirstandsecondfundamentalformsintheGauss-Weingartenequationsforspacesurfaces(see§3.9↓).AnotherusefulremarkinthiscontextisthatfromthefirstandthelastoftheFrenet-Serretformulae(seeEqs.136↓and138↓),wehave:(121)|κτ| = |T’⋅B’|wheretheprimestandsforderivativewithrespecttoanaturalparametersofthecurve.

2.3.1 CurvatureThecurvatureκofaspacecurveisameasureofhowmuchthecurvebendsasitprogressesinthetangentdirectionataparticularpoint.Thecurvaturerepresentsthemagnitudeoftherateofchangeofthedirectionofthetangentvectorwithrespecttothearclengthandhenceitisameasureforthedepartureofthecurvefromtheorientationofthestraightlinepassingthroughthatpointandorientedinthetangentdirection.Consequently,thecurvaturevanishesidenticallyforstraightlines(see§5.1↓).Infact,havinganidenticallyvanishingcurvatureisanecessaryandsufficientconditionforacurveofclassC2tobeastraightline.FromthefirstoftheFrenet-Serretformulae(Eq.136↓)andthefactthat:(122)(N⋅T)’ = (0)’ = 0 ⇒  N⋅T’ =  − N’⋅TwhichisbasedontheorthogonalityofNandTandtheproductruleofdifferentiation,thecurvatureκcanbeexpressedas:(123)κ = N⋅T’ =  − N’⋅Twheretheprimerepresentsdifferentiationwithrespecttothearclengthsofthecurve.Theminussigninthesecondequalityisconsistentwiththefactthatκisnon-negativesincethecomponentofN’inthetangentialdirectionisanti-paralleltoT.Asforthefirstequality,NandT’areparallel(seeEqs.108↑and110↑)andhencethedotproductisnon-negativeasitshouldbe.The“radiusofcurvature”,whichistheradiusoftheosculatingcircle(see§2.6↓),isdefinedateachpointofaspacecurveatwhichκ ≠ 0asthereciprocalofthecurvature,thatis:(124)Rκ = 1 ⁄ κAdifferentwayforintroducingtheseconcepts,whichisfollowedbysomeauthors,istodefinefirsttheradiusofcurvatureasthereciprocalofthemagnitudeoftheaccelerationvector,thatisRκ = (1 ⁄ | r’’(s)|)wherer(s)isthespatialrepresentationofans-parameterizedcurve;thecurvatureisthendefinedasthereciprocaloftheradiusofcurvature.Hence,theradiusofcurvaturemaybedescribedasthereciprocalofthenormoftheaccelerationvectorwhereaccelerationmeansthesecondderivativeofthespatialrepresentationofthecurvewithrespecttoitsnaturalparameter.Theremaybeanadvantageinusingtheconceptof“curvature”astheprincipalconceptinsteadof“radiusofcurvature”,thatisthecurvaturecanbedefinedatallpointsofasmoothcurvewhereatangentvectorisdefined,includingthosewithvanishingcurvature,whiletheradiusofcurvatureisdefinedonlyatthosepointswithnon-vanishingcurvature.Asindicatedabove,ifCisaspacecurveofclassC2whichisdefinedonarealintervalI ⊆ ℝandisparameterizedbyarclengths ∈ I,thatisC(s):I→ℝ3,thenthecurvatureofCatagivenpointPonthecurveisdefinedby:(125)κ = |r’’(s)|wherer(s)isthespatialrepresentationofthecurve,thedoubleprimerepresentsthesecondderivativewithrespecttos,andr’’isevaluatedatP.Foraspacecurverepresentedparametricallybyr(t),wheretisageneralparameter,wehave:(126)κ = |Ṫ| ⁄ |ṙ| = |ṙ × r| ⁄ |ṙ|3 = √([ṙ⋅ṙ][r⋅r] − [ṙ⋅r]2) ⁄ ( ṙ⋅ṙ)3 ⁄ 2whereallthequantities,whicharefunctionsoft,areevaluatedatagivenpointcorrespondingtoagivenvalueoft,andtheoverdotrepresentsderivativewithrespecttot.ItisnoteworthythatallsurfacecurvespassingthroughapointPonasurfaceSandhavethesameosculatingplaneatPhaveidenticalcurvatureκatPiftheosculatingplaneisnottangenttoSatP.

2.3.2 Torsion

Thetorsionτrepresentstherateofchangeoftheosculatingplane,andhenceitquantifiesthetwisting,inmagnitudeandsense,ofthespacecurveoutoftheplaneofcurvatureanditsdeviationfrombeingaplanecurve(see§5.2↓).Thetorsionthereforevanishesidenticallyfor

planecurves.Infact,havinganidenticallyvanishingtorsionisanecessaryandsufficientconditionforacurveofclassC2tobeaplanecurve.IfCisaspacecurveofclassC2whichisdefinedonarealintervalI ⊆ ℝanditisparameterizedbyarclengths ∈ I,thatisC(s):I→ℝ3,thenthetorsionofCatagivenpointPonthecurveisgivenby:(127)τ = N’·BwhereN’andBareevaluatedatPandtheprimerepresentsdifferentiationwithrespecttos.ThisequationcanbeobtainedfromthesecondoftheFrenet-Serretformulae(Eq.137↓)bydotproductingbothsideswithB.Theformulamayalsobegivenas:(128)τ =  − N·B’forthesamereasonasthatgivenforthealternativeformulaeofκ(seeEq.123↑andrelatedtext)orbydotproductingbothsidesofthethirdoftheFrenet-Serretformulae(Eq.138↓)withN.Foraspacecurverepresentedparametricallybyr(t),wheretisageneralparameter,wehave:(129)τ = [ṙ⋅(r × r)] ⁄ |ṙ × r|2 = [ṙ⋅(r × r)] ⁄ [(ṙ × r)⋅(ṙ × r)] = [ṙ⋅(r × r)] ⁄ [(ṙ⋅ṙ)(r⋅r) − (ṙ⋅r)2]whereallthequantities,whicharefunctionsoft,areevaluatedatagivenpointPcorrespondingtoagivenvalueoft,andtheoverdotrepresentsderivativewithrespecttot.Thecurveshouldhavenon-vanishingcurvatureκatP.ForrectangularCartesiancoordinates,thetorsionofans-parameterizedcurveisgivenintensornotationby:(130)τ = (ϵijkxi’xj’’xk’’’) ⁄ κ2whereκisthecurvatureofthecurveasdefinedpreviously.Thelastformulaisbasedonitspredecessor.Forgeneralcurvilinearcoordinates,thetorsionofans-parameterizedcurveisgivenintensornotationby:(131)τ = εijkTiNj(δNk ⁄ δs)Themagnitudeoftorsionisindependentofthenatureofthecurveparameterizationandorientationasdeterminedbythesenseofincreaseofitsparameter.Itisalsoinvariantunderpermissiblecoordinatetransformations.Finally,the“radiusoftorsion”isdefinedateachpointofaspacecurveforwhichτ ≠ 0astheabsolutevalueofthereciprocalofthetorsion,thatis:(132)Rτ = |1 ⁄ τ|Wenotethatsomeauthorsdonottaketheabsolutevalueandaccordinglytheradiusoftorsioncanbenegative.

2.4 GeodesicTorsionGeodesictorsion,whichisalsoknownastherelativetorsion,isanattributeofacurveembeddedinasurface.ThegeodesictorsionofasurfacecurveCatagivenpointPisthetorsionofthegeodesiccurve(see§5.7↓)thatpassesthroughPinthetangentdirectionofCatP.Aswewillseein§5.7↓,intheneighborhoodofagivenpointPonasmoothsurfaceandforanyspecifieddirectionthereisoneandonlyonegeodesiccurvepassingthroughPinthatdirection.Thegeodesictorsionτgofasurfacecurverepresentedspatiallybyr(s)isgivenbythefollowingscalartripleproduct:(133)τg = n⋅(n’ × r’)wherenistheunitnormalvectortothesurface,theprimesrepresentdifferentiationwithrespecttothenaturalparameters,andallthesequantitiesareevaluatedatagivenpointonthecurvecorrespondingtoagivenvalueofs.ThegeodesictorsionofacurveCatanon-umbilicalpointP(see§4.10↓)isgivenintermsoftheprincipalcurvaturesκ1andκ2(see§4.4↓)by:(134)τg = (κ1 − κ2)sinθcosθwhereθistheanglebetweenthetangentvectorTtothecurveCatPandthefirstprincipaldirectiond1(seeDarbouxframein§4.4↓).ThegeodesictorsionofasurfacecurveCparameterizedbyarclengthsatagivenpointPisalsogivenintermsofthecurvetorsionby:(135)τg = τ − (dφ ⁄ ds)whereτisthetorsionofCatP,andφistheanglebetweentheunitnormalvectorntothesurfaceandtheprincipalnormalvectorNofCatP,i.e.φ = arccos(n⋅N).Also,thecurveCshouldnotbeasymptotic(see§5.9↓).Thisformula(Eq.135↑)whichisknownastheBonnetformula,demonstratesthatwhennandNarecollinearalongthecurve,thegeodesictorsionandthetorsionareequal(i.e.τg = τ).Aswewillseein§5.7↓,whennandNarecollinear,thegeodesiccomponentofthecurvaturevector(see§4.1↓)willvanish.Inthiscase,thegeodesiccurvature(see§4.3↓)willvanishandthecurvebecomesageodesic.Soinbrief,onageodesiccurvewehave:τg = τwhichisconsistentwiththeabovestatementatthebeginningofthissection.ThegeodesictorsionofasurfacecurveCatagivenpointPiszeroiffCistangenttoalineofcurvatureatP(see§5.8↓).Hence,onalineofcurvaturethegeodesictorsionvanishesidentically.ThiscanbeseenfromEq.134↑whereeithersinθorcosθvanishes.Thegeodesictorsionsoftwoorthogonalsurfacecurvesattheirpointofintersectionareequalinmagnitudeandoppositeinsign.

2.5 RelationshipbetweenCurveBasisVectorsandtheirDerivativesThethreebasisvectorsT, N, BofaspacecurveareconnectedtotheirderivativesbytheFrenet-SerretformulaewhicharegiveninrectangularCartesiancoordinatesby:(136)dTi ⁄ ds = κNi

(137)dNi ⁄ ds = τBi − κTi(138)dBi ⁄ ds =  − τNi

Asindicatedpreviously(see§2.2↑),thesignofthetermsinvolvingτdependsontheconventionaboutthetorsionandhencetheseequationsdifferbetweendifferentauthors.TheaboveequationsarealsoknownasFrenetformulae.TheFrenet-Serretformulaecanbecastinthefollowingmatrixformusingsymbolicnotation:(139)

whereallthequantitiesinthisequationarefunctionsofarclengthsandtheprimerepresentsderivativewithrespecttos.Asseen,thecoefficientmatrixofthissystemisanti-symmetric.TheFrenet-Serretformulaecanalsobegiveninthefollowingform:(140)T’ = d × T(141)N’ = d × N(142)B’ = d × Bwheredisthe“Darbouxvector”whichisgivenby:(143)d = τT + κBThisformoftheFrenet-Serretformulaeismorememorableapartfromtheexpressionofd.WenotethatsomeauthorsdefinedasascalarmultipleofwhatisgiveninEq.143↑.Theabovethreeequations(i.e.Eqs.140↑-142↑)maybemergedinasingleequationas:(144)(T’, N’, B’) = d × (T, N, B)Ingeneralcurvilinearcoordinates,theFrenet-Serretformulaearegivenintermsoftheabsolutederivativesofthethreevectorsby:(145)δTi ⁄ δs = (dTi ⁄ ds) + ΓijkTj(dxk ⁄ ds) = κNi

(146)δNi ⁄ δs = (dNi ⁄ ds) + ΓijkNj(dxk ⁄ ds) = τBi − κTi

(147)δBi ⁄ δs = (dBi ⁄ ds) + ΓijkBj(dxk ⁄ ds) =  − τNi

wheretheindexedxrepresentgeneralspatialcoordinatesandsisanaturalparameterwhiletheothersymbolsareasdefinedearlier.Accordingtothefundamentaltheoremofspacecurves,whichisoutlinedpreviouslyin§2.3↑,acurvedoesexistanditisuniqueiffitscurvatureandtorsionasfunctionsofarclengtharegiven.Now,itisnaturaltoexpectthatsuchasolutioncanbeobtainedfromthesystemofdifferentialequationsgivenbytheFrenet-Serretformulae.However,suchasolutioncannotbeobtainedingeneralbydirectintegrationoftheseequations.Moreelaboratemethods(e.g.methodsbasedontheRiccatiequationforreducingasystemofsimultaneousdifferentialequationstoafirstorderdifferentialequation)maybeusedtoobtainthesolution.Nevertheless,asolutioncanbeobtainedbydirectintegrationoftheFrenet-Serretformulaeforplanecurves(see§5.2↓)wherethetorsionvanishesidentically.AsolutionbydirectintegrationoftheFrenet-Serretformulaecanalsobeobtainedinothersimplecasessuchaswhenthecurvatureandtorsionareconstants.

2.6 OsculatingCircleandSphereAtanypointPwithnon-zerocurvatureofasmoothspacecurveC,an“osculatingcircle”(Fig.25↓),whichmayalsobecalledthecircleofcurvatureorthekissingcircle,canbedefinedwherethiscircleischaracterizedby:1.ItistangenttoCatP,i.e.thecircleandthecurvehaveacommontangentvectoratP.2.ItliesintheosculatingplaneofCatP.3.ItsradiusRκisequalto(1 ⁄ κ)whereκisthecurvatureofCatP.4.ItscenterCcislocatedatrCwhichisgivenby:(148)rC = rP + (1 ⁄ κ)N = rP + RκNwhererPisthepositionvectorofPandNistheprincipalnormalvectorofCatP.Thecenterofcurvatureofacurveatapointonthecurveisdefinedasthecenteroftheosculatingcircleatthatpoint,asgivenabove.IfthecurveCisacircle,thenthecenterofcurvatureatanypointisthecenterofthecircleitself,sothecircleisitsownosculatingcircle.

Figure25 TheosculatingcircleCoofaspacecurveCatpointPwiththeprincipalnormalvectorN,centerofcurvatureCcandradiusofcurvatureRκ.Theosculatingcircleprovidesagoodapproximationtothecurveintheneighborhoodofitspointswheretheosculatingcircleisdefined.Followingthemannerofdefiningthetangentlinetoacurveasalimitofthesecantline(see§2.1↑),theosculatingcircletoacurveatagivenpointPmaybedefinedgeometricallyasthelimitofacirclepassingthroughPandtwootherpointsonthecurveasthesetwopointsconvergetoPwhilestayingonthecurve(Fig26↓).Itshouldberemarkedthatinsomecasestheosculatingcircleanditsparametersmaybedefinedgeometricallybutnotanalyticallywhenthesecondderivativeofthecurveatthegivenpointisnotproperlydefinedtodeterminetheradiusofcurvature(seeEq.§125↑).

Figure26 Theosculatingcircle(small)ofacurveCatapointPasalimitofanothercircle(big)passingthroughPandtwootherpoints,P1andP2.Followingthemannerofdefiningtheosculatingcircleasalimit,the“osculatingsphere”ofacurveCatagivenpointPmaybedefinedsimilarlyasthelimitofaspherepassingthroughPandthreeneighboringpointsonthecurveasthesethreepointsconvergetoP.ThepositionofthecenterCSoftheosculatingsphereatP,whichiscalledthecenterofsphericalcurvatureofCatP,isgivenby:(149)rS = rP + (1 ⁄ κ)N − [κ’ ⁄ (τκ2)] B = rP + RκN + sgn(τ)RτRκ’BwhererSandrParethepositionvectorsofCSandP,BandNarethebinormalandprincipalnormalvectors,κandτarethecurvatureandtorsion,RκandRτaretheradiiofcurvatureandtorsion,sgn(τ)isthesignfunctionofτ(s),andtheprimerepresentsderivativewithrespecttoanaturalparametersofC.AllthesequantitiesbelongtoCatPwhichshouldhavenon-vanishingcurvatureandtorsion,i.e.κ, τ ≠ 0.FromEq.149↑,itcanbeseenthattheradiusoftheosculatingsphereisgivenby:(150)|rS − rP| = √(R2κ + [RτRκ’]2)

2.7 ParallelismandParallelPropagationInflatspaces,parallelismisanabsolutepropertyasitisdefinedwithoutreferencetoaperipheralobject.However,inRiemannianspacestheideaofparallelismisdefinedinreferencetoaprescribedcurveandhenceitisdifferentfromtheideaofparallelismintheEuclideansense.AvectorfieldAαisdescribedasbeingparallelalongthesurfacecurveuβ = uβ(t)iffitsabsolutederivative(see§7↓)alongthecurvevanishes,thatis:(151)δAα ⁄ δt ≡ Aα;β(duβ ⁄ dt) ≡ 

(dAα ⁄ dt) + ΓαβγAγ(duβ ⁄ dt) = 0Thismeansthatthesufficientandnecessaryconditionforavectorfieldtobeparallelalongasurfacecurveisthatthecovariantderivativeofthefieldisnormaltothesurface.Allthevectorsofafieldofparallelvectorshavethesameconstantmagnitude.Afieldofabsolutelyparallelunitvectorsonasurfacedoexistiffthereisanisometriccorrespondencebetweentheplaneandthesurface.WhentwosurfacesaretangenttoeachotheralongagivencurveC,thenavectorfieldwhichisparallelalongCwithrespecttooneofthesesurfaceswillalsobeparallelalongCwithrespecttotheothersurface.Whentwonon-trivialvectorsexperienceparallelpropagationalongaparticularcurvetheanglebetweenthemstaysconstant.AsaconsequenceofthedefinitionofparallelisminRiemannianspacesandthepreviousstatements,wehave:1.Parallelpropagationispathdependent.Hence,asurfacevectorfieldparallellypropagatedalongagivencurvebetweentwopointsP1andP2onthecurvedoesnotnecessarilycoincidewithanothervectorfieldparallellypropagatedalonganothercurveconnectingP1andP2.2.Sinceparallelpropagationdependsonthepathofpropagation,thengiventwopointsP1andP2onasurface,thevectorobtainedatP2byparallelpropagationofavectorfromP1alongagivensurfacecurveCconnectingP1toP2dependsonthecurveC.3.StartingfromagivenpointPonaclosedsurfacecurveCenclosingasimplyconnectedregion(see§3.1↓)onthesurface,parallelpropagationofavectorfieldaroundCstartingfromPdoesnotnecessarilyresultinthesamevectorfieldwhenarrivingatP.WenotethattheanglebetweentheinitialandfinalvectorsinthissituationisameasureoftheGaussiancurvatureonthesurface(see§4.5↓).4.IfC:I → SisaregularcurveonasurfaceSdefinedontheintervalI ⊆ ℝ,andv1andv2areparallelvectorfieldsoverC,thenthedotproductv1⋅ v2whichisassociatedwiththemetrictensor,thenormofthevectorfields|v1|and|v2|,andtheanglebetweenv1andv2areconstants.

2.8 ExercisesExercise1.Statethetechnicaldefinitionofspacecurveoutliningthedifferencebetweenacurveanditstrace.Exercise2.Whatisthemostcommonwayofdefiningspacecurvesmathematically?Giveanexamplefromsimplecurveslikecircleandellipse.Exercise3.Makeacleardistinctionbetweengeneralandnaturalparameterizationofspacecurves.Exercise4.Stateamathematicalconditionforaspacecurvetobeparameterizednaturally.Exercise5.Whatistherelationbetweentwonaturalparametersofagivenspacecurve?Exercise6.Thefollowingequation:r(t) = (t ⁄ 2,  − t ⁄ 2, t ⁄ √(2))isaparametricrepresentationofaspacecurve.Istanaturalparameterornot?Justifyyouranswer.Exercise7.Provethattwonaturalparameters,sandš,ofacurvearerelatedbytheequationš = ±s + cwherecisarealconstant.Exercise8.Usingtensornotation,writedowntheequationofthetangentvectortoasurfacecurverepresentedparametricallybyC(u1(t), u2(t))wheretisageneralparameter.Exercise9.Whatisthemeaningofhaving“regularcurveataspecificpoint”?What“regularcurve”means?Exercise10.Provethattheparametricrepresentation:r(t) = (t2, et, t + 1)ofaspacecurveisregularforallt.Exercise11.Statetheconditionforavectortobetangenttoaregularsurfaceatagivenpointonthesurface.Exercise12.Findtheunittangentvector,T(t),foraspacecurverepresentedby:r(t) = (t2, t, sint).Exercise13.Findthearclengthofaspacecurvegivenby:r(t) = (5t, 7cosht, 2sinht)for1 ≤ t ≤ 4.Exercise14.Findthecurvature,asafunctionoft,ofaspacecurverepresentedby:r(t) = (cost − 1, sint + t, t2).Exercise15.Define“periodiccurve”givingtwocommonexamplesotherthanthosegiveninthetext.Exercise16.Shouldaperiodiccurvebeaplanecurve?Exercise17.Shouldaperiodiccontinuouscurvebeaclosedcurve?Shouldaperiodicsmoothcurvebeaclosedcurve?Exercise18.AplanecurvecalledcissoidofDioclesisgiveninpolarcoordinatesby:ρ = 2sinφtanφ.FindtheequationofthecurveinarectangularCartesiancoordinatesystem.Exercise19.Sketchthecurveofthepreviousquestionfor0 ≤ |φ| ≤ (π ⁄ 4)(noticethetwobranches).Exercise20.Showthatforacurverepresentedspatiallybyrandparameterizednaturallybysandgenerallybyt,therelationbetweensandtisgivenby:|ds ⁄ dt| = |dr ⁄ dt|.Exercise21.DefineFrenetframewithasimplesketchshowingthebasisvectorsatagivenpointonanarbitraryspacecurve.Exercise22.WritedownthemathematicalequationsoftheunitvectorsT, N, Bforacurveparameterizedbyageneralparametertandanaturalparameters.Exercise23.Statethemathematicaldefinitionofthecurvatureκandthetorsionτofans-parameterizedspacecurve.Exercise24.Showthatthesufficientandnecessaryconditionforaspacecurvetobeaplanecurveisthatitstorsionvanishesidentically.Exercise25.Whatarethecurvatureandtorsionof(a)astraightline(b)acirclewithradiusR = 3.2(c)acurveparameterizedby:r(t) = (3cos(t), 3sin(t), 1.9t)?Exercise26.Findthetorsion,asafunctionoft,of(a)acurverepresentedby:r(t) = (sint + t, cost − 3, t + 2)(b)acurverepresentedby:r(t) = (t, 2t2, t3)

(c)acurverepresentedby:r(t) = (at, bt3, ct2)wherea, b, carenon-vanishingrealconstants.Exercise27.WhatarethemathematicalconditionsthatrepresentthefactthatthevectorsT, N, Baremutuallyorthogonalandofunitlength?Exercise28.ProvethetheoremofLancretwhichstatesthataspacecurveofclassC3withnon-vanishingcurvatureisahelixifftheratioofitstorsiontoitscurvatureisconstantalongthecurve.Exercise29.Showthatiftwospacecurves,whichhaveaninjectiveassociation,possessparalleltangentvectorsattheircorrespondingpoints,thentheirnormalandbinormalvectorsatthesepointsareparallelaswell.Exercise30.ProveEq.121↑(i.e.|κτ| = |T’⋅B’|).Exercise31.Giveaparametricrepresentationofacircularhelixusinganaturalparameters.Exercise32.Foraplanecurveina3Dspacegivenbytheequation:y = 2x2 − x + 3(0 ≤ x ≤ 10),findtheequationsoftheosculating,normalandrectifyingplanesatpoint(1, 4).Exercise33.Foraspacecurveparameterizedas:(x, y, z) = (t, t3, 3t2),findtheequationsofthetangent,principalnormalandbinormallinespassingthroughthepoint(1, 1, 3)onthecurve.Exercise34.Giveanexampleofanon-planarspacecurvewhoseprincipalnormalvectorsatallpointsofthecurveareparalleltoaparticularplane.Exercise35.Foraspacecurveparameterizedas:(x, y, z) = (cost, sint, 5t),findtheequationsoftheosculating,rectifyingandnormalplanesatthepointonthecurvewitht = 1.3.Exercise36.WhichofthethreevectorsT, N, Bisnotaffectedbyreversingthesenseoftraversingthespacecurve?Exercise37.WhataretheprincipalnormalvectorNandthebinormalvectorBofahelixrepresentedby:r(t) = (cos3t, sin3t, 3t)?Exercise38.FindthethreevectorsT, N, Balongacurverepresentedby:r(t) = (2t − 3, t3 + t, 5 − t2).Exercise39.Makeasimplesketchfortheosculating,rectifyingandnormalplanesatapointonanarbitraryspacecurve.Useacomputergraphicpackageifconvenient.Exercise40.WritedowntheequationofLancretrelatedtothethirdcurvatureofspacecurvesanddiscussitssignificance.Exercise41.Discuss,indetail,thefundamentaltheoremofspacecurvesindifferentialgeometryoutliningitsapplicationandsignificance.Exercise42.Giventhatthecurvatureofaplanecurveisgivenby:κ = 1 ⁄ (3s + 5)wheres > 0isanaturalparameter,findtheequationofthiscurve.Exercise43.WritedowntheFrenet-SerretformulaeofaspacecurveinarectangularCartesiancoordinatesystemexplainingallthesymbolsinvolved.Exercise44.ByintegratingtheFrenet-Serretformulae,obtainthesolutionofaspacecurvewithκ = aandτ = bwherea, b > 0arerealconstants.Exercise45.Identifythetypeofthecurveinthepreviousquestion.Exercise46.Whatarethe“intrinsic”or“natural”equationsofacurve?Exercise47.FindtheintrinsicequationsofthecatenarydefinedbyEqs.28↑-29↑.Exercise48.Findtheparametricrepresentationofacurvewiththefollowingintrinsicequations:κ = √(c ⁄ s)andτ = 0wherec > 0isarealconstantands > 0isanaturalparameter.Exercise49.Provethatthecurvatureofat-parameterizedspacecurveisgivenby:κ = |ṙ × r| ⁄ |ṙ|3Exercise50.Provethatthetorsionofat-parameterizedspacecurveisgivenby:τ = [ṙ⋅(r × r)] ⁄ |ṙ × r|2Exercise51.Whichofthetwomaincurveparameters,κandτ,isnecessarilynon-negativeandwhy?Exercise52.Showthatalongans-parameterizedcurver(s),thefollowingrelationholdstrue:r’⋅(r’’ × r’’’) = τκ2.Exercise53.Canweobtainthecurvatureandtorsionofcircleasaspecialcaseofthecurvatureandtorsionofhelix?Ifyes,how?Isthisconsistentwiththedefinitionofhelixasgivenin§1.4.1↑(seeEqs.4↑-6↑andFig.2↑)?Exercise54.Findthecurvatureandtorsionofaspacecurverepresentedby:

r(t) = (t3, t + 1,  − t2)atthepointwitht = 2.4.Exercise55.Giveanexampleofaspacecurvewhosecurvatureandtorsionarevariables.Exercise56.Investigatetherelationbetweenthecurvatureandtorsionatcorrespondingpointsoftwospacecurveswhicharemirror-reflectionofeachotherwithrespecttoagivenplane.Exercise57.Investigatetherelationbetweenthecurvatureandtorsionatcorrespondingpointsoftwospacecurveswhicharesymmetricwithrespecttoagivenpoint.Exercise58.Discuss,indetail,theconceptof“1Dinhabitant”inthecontextofclassifyingthepropertiesofspacecurves.Exercise59.Establishacorrespondencebetweenthetwomainparametersofspacecurve(i.e.curvatureandtorsion)andthefirstandsecondfundamentalformsofspacesurface.Exercise60.Discusstheresemblancebetweenκandτofspacecurveandthecurvaturetensorofspacesurface.Exercise61.State,inwords,themathematicalrelation:τ = εijkTiNj(δNk ⁄ δs)usingtechnicaltermsforallthenotationsandsymbolsusedinthisequation.Exercise62.Whatisthesignificanceofthecurvatureandtorsionofspacecurvesasmeasuresoftheirvariationintheembeddingspace?Exercise63.Whatistherelationbetweenthecurvatureandtheradiusofcurvatureofaspacecurve?Isthereanadvantageinusingoneoftheseortheotherasthemainconcept?Exercise64.Outlinethephysicalsignificanceoftherelation:κ = |r’’|.Exercise65.Expressκintermsofranditsfirstandsecondderivativeswhererisaspatialrepresentationofacurveparameterizedbyageneralparametert.Exercise66.ExpressτintermsofN’andBofanaturallyparameterizedcurve.Exercise67.Expressτintermsofranditsfirst,secondandthirdderivativeswhererisaspatialrepresentationofacurveparameterizedbyageneralparametert.Exercise68.Whatistherelationbetweenthetorsionandtheradiusoftorsionofaspacecurve?Exercise69.Definegeodesictorsioninwordsandstateitsmathematicalrelationtorandnandtheirderivatives.Exercise70.Whatisthesignificanceoftherelation:τg = τ − (dφ ⁄ ds)andwhatthesymbolsinthisrelationmean?Exercise71.Whatistheconditionforthetorsionandthegeodesictorsionofaspacecurvetobeequal?Exercise72.Provethatalongasufficientlysmoothcurverepresentedbyr(t),thevectorratagivenpointPonthecurveisparalleltotheosculatingplaneatP.Exercise73.Obtaintheequationoftheosculatingplaneofacurverepresentedparametricallyby:r(t) = (3cost, 2sint, cost + 5sint)atageneralpointonthecurve.Exercise74.DerivethesecondoftheFrenet-Serretformulaeusingtheothertwoformulae.Exercise75.DefineDarbouxvectordandhenceverifythattherelationsgivenbyEqs.140↑-142↑arevalid.Exercise76.Explainallthesymbolsandnotationsusedinthefollowingrelation:τ = [ṙ⋅(r × r)] ⁄ [(ṙ⋅ṙ)(r⋅r) − (ṙ⋅r)2]Exercise77.Provethatforacurvewithhelicalshape,theDarbouxvectorisconstantalongthecurve.Exercise78.StatetheFrenet-SerretformulaeinasingleequationusingtheDarbouxvector.Exercise79.WritedowntheFrenet-Serretformulaeassumingageneralcurvilinearcoordinatesystem.Exercise80.GiveabriefexplanationofwhythesolutionofaspacecurvecannotbeobtainedingeneralbyadirectintegrationoftheFrenet-Serretequations.Exercise81.Giveabriefdefinitionoftheosculatingcircleandtheosculatingsphereofaspacecurve.Exercise82.Howtheosculatingcircleandtheosculatingsphereofaspacecurvecanbedefinedusingtheconceptoflimit?Exercise83.ProvethatforagivenspacecurveC,thebinormallinesofCandthetangentlinestothelocusofthecentersofsphericalcurvatureofCareparallelattheircorrespondingpoints.Exercise84.Deriveanexpressionforthepositionofthecenterofcurvatureofat-parameterizedtwistedcurverepresentedspatiallybyr(t)intermsofranditsderivatives.

Exercise85.Findthespatialcoordinatesofthecenteroftheosculatingcircleofaspacecurverepresentedby:r(t) = (t, √(t), t2)atthepointwitht = 2.6.Exercise86.Whatistherelationbetweentheosculatingcircleandtheosculatingplaneatagivenpointofaspacecurve?Exercise87.Explainallthesymbolsinvolvedintheequation:rS = rP + RκN + sgn(τ)RτRκ’Bwhichisrelatedtothecenteroftheosculatingsphere.Exercise88.Writedowntheformulafortheradiusoftheosculatingsphereexplainingallthesymbolsused.Exercise89.WhatisthedifferencebetweentheconceptofparallelisminEuclideanandnon-Euclideanspaces?Exercise90.Statethemathematicalconditionforavectorfieldtobeparallelalongagivensurfacecurveintermsofitsabsolutederivative.Exercise91.Whatisthemeaningofdescribingparallelpropagationaspathdependent?Whataretheconsequencesofthisdependency?

Chapter3SurfacesinSpaceHere,weexaminesufficientlysmoothsurfacesembeddedina3DEuclideanspaceusingaCartesiancoordinatesystem(x, y, z)forthemostparts.SomepartsarebasedonamoregeneralRiemannianspacewithacurvilinearcoordinatesystem.

3.1 GeneralBackgroundaboutSurfacesA2Dsurfaceembeddedina3Dspacemaybedefinedlooselyasasetofconnectedpointsinthespacesuchthattheimmediateneighborhoodofeachpointonthesurfacecanbedeformedcontinuouslytoformaflatdisk.Technically,asurfaceina3Dmanifoldisamappingfromasubsetoftheparametersplanetoa3Dspace,thatisS:Ω → ℝ3,whereΩisasubsetofℝ2planeandSisasufficientlysmoothinjectivefunction.Similarconditionsmayalsobeimposedtoensuretheexistenceofatangentplaneandanormalvectorateachpointofthesurface.Inparticular,thecondition∂ur × ∂vr ≠ 0atallpointsonthesurfaceisusuallyimposedtoensureregularityofthesurface.Likespaceandsurfacecurves,theimageofthemappinginℝ3isknownasthetraceofthesurface.Forconvenience,weusecurveandsurfaceinthepresentbookfortraceaswellasformapping;themeaningshouldbeobviousfromthecontext.Weshouldremarkthatthetraceofacurveorasurfaceshouldnotbeconfusedwiththetraceofamatrixwhichisthesumofitsdiagonalelements.A2Dsurfaceembeddedina3Dspacecanbedefinedexplicitlyby:z = f(x, y),orimplicitlyby:F(x, y, z) = 0,orparametricallyby:x(u1, u2), y(u1, u2), z(u1, u2)whereu1andu2(oruandv)arethesurfacecoordinatesontheparametersplane,asdefinedpreviously(see§1.4.3↑),whicharemutuallyindependentparameters.Bysubstitution,eliminationandalgebraicmanipulationtheseformscanbetransformedinterchangeably.Acoordinatepatchofasurfaceisaninjective,bicontinuous,regular,parametricrepresentationofapartofthesurface.Inmoretechnicalterms,acoordinatepatchofclassCn(n > 0)onaspacesurfaceSisafunctionalmappingofanopensetΩintheuvparametersplaneontoSthatsatisfiesthefollowingconditions:1.ThefunctionalmappingrelationisofclassCnoverΩ.2.Themappingisone-to-oneandbicontinuousoverΩ.3.E1 ×  E2 ≠  0atanypointinΩwhereE1andE2arethesurfacebasisvectors(see§1.4.5↑and1.4.3↑).Asindicatedpreviously,avectorvisdescribedasatangentvectortothesurfaceSatagivenpointPonthesurfaceifthereisaregularcurveCembeddedinSandpassingthroughPsuchthatvisatangenttothecurveatP,i.e.v = c(dr ⁄ dt)(c ∈ ℝ, c ≠ 0)whererisat-parameterizedpositionvectorrepresentingC.ThesetofalltangentvectorstothesurfaceSatpointPformsatangentplanetoSatP(Fig.27↓).ThissetiscalledthetangentspaceofSatPanditisusuallynotatedwithTPS.Itisobviousthatthereexistinfinitelymanytangentvectors,withvaryingmagnitudeanddirection,toasmoothsurfaceatitsregularpoints.

Figure27 Tangentplane(solid)ofasurface(outlinedbyagrid)atagivenpointalongsidethesurfacebasisvectors,E1andE2,andtheunitnormalvector,n,atthatpoint.Aswewillsee(alsoreferto§1.4.5↑),thetangentspaceofaregularsurfaceatagivenpointonthesurfaceisthespanofthetwolinearlyindependentbasisvectorsdefinedas:(152)E1 = ∂ r ⁄ ∂u1E2 = ∂ r ⁄ ∂u2wherer(u1, u2)isthespatialrepresentationofthesurfacecoordinatecurvesandu1andu2arethesurfacecoordinates,asexplainedbefore.Thetangentspace,therefore,istheplanepassingthroughPandisperpendiculartothevectorE1 ×  E2.Asindicatedpreviously,everyvectortangenttoaregularsurfaceSatagivenpointPonScanbeexpressedasalinearcombinationofthesurfacebasisvectorsE1andE2atP.Thereverseisalsotrue,thatiseverylinearcombinationofE1andE2atPisatangentvectortoaregularcurveembeddedinSandpassingthroughPandhenceitisatangenttoSatP.Basedonthepreviousstatements,weseethatthetangentplaneofasurfaceatagivenpointPonthesurfacecanbegivenby:

(153)r = rP + pE1 + qE2whereristhepositionvectorofanarbitrarypointonthetangentplane,rPisthepositionvectorofthepointP,p, q ∈ ( − ∞, ∞)arerealvariables,andE1andE2arethesurfacebasisvectorsatP.Alternatively,thetangentplanecanbeexpressedintermsofthenormalvectorntothesurfaceatPby:(154)(r − rP)⋅n = 0ThetangentspaceataspecificpointPofasurfaceisapropertyofthesurfaceatPandhenceitisindependentofthepatchthatcontainsPandtheparticularparameterizationofthesurface.Foranynon-trivialvectorvwhichisparalleltothetangentplaneofasimpleandsmoothsurfaceSatagivenpointPonS,thereisacurveinSpassingthroughPandrepresentedparametricallybyr(t)suchthatv = c(dr ⁄ dt)wherec ≠ 0isarealconstant.Asaresultofthelastandthepreviousstatements,anon-trivialvectorisparalleltothetangentplaneofasurfaceSatagivenpointPiffitisequaltoatangentvectortoSatP.ThestraightlinepassingthroughagivenpointPonasurfaceSinthedirectionofthenormalvectorofSatPiscalledthenormallinetoSatP.Theequationofthisnormallineisgivenby:(155)r = rP + knwhereristhepositionvectorofanarbitrarypointonthenormalline,rPisthepositionvectorofthepointP,k ∈ ( − ∞, ∞)isarealvariable,andnistheunitnormalvectortothesurfaceatP.WeremarkthatthisnormallineshouldnotbeconfusedwiththenormallineofasurfacecurveCthatpassesthroughPwhichisusuallycalledtheprincipalnormallineofC(see§2.2↑).Anyway,thetwoshouldbeeasilydistinguishedbynoticingtheiraffiliationtosurfaceorcurve.Asstatedbefore,asurfaceisregularatagivenpointPiffE1 ×  E2 ≠  0atPwhereE1 = ∂urandE2 = ∂vrarethetangentvectorstothecoordinatecurvesatP.AsurfaceisregulariffE1 ×  E2 ≠  0atanypointonthesurface.Aregularcurve(see§2.1↑)ofclassCnonasufficientlysmoothsurfaceisanimageofauniqueregularplanecurveofclassCnintheparametersplane,whereinthisstatementweareconsideringeachconnectedpartofthecurvebeingembeddedinacoordinatepatchifthereisnosinglepatchthatcontainstheentirecurve.A“Mongepatch”isacoordinatepatchina3Dspacedefinedbyafunctioninoneofthefollowingthreeforms:(156)r(u, v) = (f(u, v), u, v)(157)r(u, v) = (u, f(u, v), v)(158)r(u, v) = (u, v, f(u, v))wherefisadifferentiablefunctionofthesurfacecoordinatesuandv.WhenfisofclassCnthenthecoordinatepatchisofthisclass.Asimplyconnectedregiononasurfacemeansthataclosedcurvecontainedintheregioncanbeshrunkdowncontinuouslyontoanypointintheregionwithoutleavingtheregion.Inmoresimpleterms,itmeansthattheregioncontainsnoholesorgapsthatseparateitsparts.Asimplesurfaceisacontinuouslydeformedplanebycompression,stretchingandbending(seeFootnote10in§8↓).Examplesofsimplesurfacearecylinders,conesandellipticandhyperbolicparaboloids.AconnectedsurfaceSisasimplesurfacewhichcannotbeentirelyrepresentedbytheunionoftwodisjointopenpointsetsinℝ3wherethesesetshavenon-emptyintersectionwithS.Hence,foranytwoarbitrarypoints,P1andP2,onSthereisacontinuouscurvewhichistotallyembeddedinSwithP1andP2beingitsendpoints.Examplesofconnectedsurfaceareplanes,ellipsoidsandcylinders.Aclosedsurfaceisasimplesurfacewithnoopenedges.Examplesofclosedsurfacearespheresandellipsoids.Aboundedsurfaceisasurfacethatcanbecontainedentirelyinasphereofafiniteradiussuchasellipsoidandtorus.AcompactsurfaceisasurfacewhichisboundedandclosedlikeatorusoraKleinbottle(Fig.28↓).IffisadifferentiableregularmappingfromasurfaceStoasurfaceS,thenifSiscompactthenSiscompact.IfS1andS2aretwosimplesurfaceswhereS1isconnectedandS2isclosedandcontainedinS1,thenthetwosurfacesareequalaspointsets.Asaresult,asimpleclosedsurfacecannotbeapropersubsetofasimpleconnectedsurface.

Figure28 Kleinbottle.Anorientablesurfaceisasimplesurfaceoverwhichacontinuously-varyingnormalvectorcanbedefined.Hence,spheres,cylindersandtoriare

orientablesurfaceswhiletheMobiusstrip(Fig.1↑)isanon-orientablesurfacesinceanormalvectormovedcontinuouslyaroundthestripfromagivenpointwillreturn,followingacompleteround,tothepointintheoppositedirection.Similarly,Kleinbottle(Fig.28↑)isanotherexampleofanon-orientablesurface.Anorientedsurfaceisanorientablesurfaceoverwhichthedirectionofthenormalvectorisdetermined.Anorientablesurfacewhichisconnectedcanbeorientedinonlyoneoftwopossibleways.Anelementarysurfaceisasimplesurfacewhichpossessesasinglecoordinatepatchbasis,andhenceitisanorientablesurfacewhichcanbemappedbicontinuouslytoanopensetintheplane.Examplesofelementarysurfaceareplanes,conesandellipticparaboloids.Adevelopablesurfaceisasurfacethatcanbeflattenedintoaplanebyunfoldingwithoutlocaldistortionbycompressionorstretching.Itiscalleddevelopablebecauseitcanbedevelopedintoaplanebyrollingthesurfaceoutonaplanewithoutcompressionorstretching.Acharacteristicfeatureofdevelopablesurfaceisthat,liketheplane,itsRiemann-Christoffelcurvaturetensor(see§1.4.10↑)andGaussiancurvature(see§4.5↓)arezeroateverypointonthesurface.Cylindersandconesareobviousexamplesofdevelopablesurface.Atopologicalpropertyofasurfaceisapropertywhichisinvariantwithrespecttoinjectivebicontinuousmappings.Anexampleofatopologicalpropertyiscompactness.AdifferentiableregularmappingfromasurfaceStoasurfaceSiscalledconformalifitpreservesanglesbetweenorientedintersectingcurvesonthesurface.Themappingisdescribedasdirectifitpreservesthesenseoftheanglesandinverseifitreversesit.Technically,themappingisconformalifthereisafunctionq(u, v) > 0thatappliestoallpatchesonthesurfacesuchthat:(159)aαβ = qãαβwhereα, β = 1, 2andtheunbarredandbarredindexedaarethecoefficientsofthesurfacecovariantmetrictensorinSandSrespectively.Theaboveconditionofconformalmappingmaybestatedbysayingthatthecoefficientsofthefirstfundamentalformsofthetwosurfacesareproportionalattheircorrespondingpoints.Anexampleofconformalmappingisthestereographicprojection(Fig.29↓)fromtheRiemannspheretoaplane.Weremarkthatstereographicprojectionisamappingoftheunitsphereontoaplanewhereeachpointofthesphereisprojected,throughthelineconnectingthispointtothenorthpoleofthesphere,ontothepointofintersectionofthatlinewiththeplane.Thisplaneisthetangentplanetothesphereatitssouthpole.

Figure29 StereographicprojectionwherethepointsP1andP2ontheunitsphereareprojectedrespectivelyonthepointsP1andP2ontheplanewithNrepresentingthenorthpoleofthesphere.Theplaneistouchingthesphereatitssouthpole.Anisometryorisometricmappingoftwosurfacesisaone-to-onemappingfromasurfaceStoasurfaceSthatpreservesdistances.Hence,anyarcinSismappedontoanarcinSwithequallength.ThetwosurfacesSandSaredescribedasisometricsurfaces.Anexampleofisometricmappingisthedeformationofarectangularplanesheetintoacylinderwithnolocaldistortionbycompressionorstretchingandhencethetwosurfacesareisometricsincealldistancesarepreservedduringthisprocess.Isometryisasymmetricrelationandhencetheinverseofanisometricmappingisanisometricmapping,thatisiffisanisometryfromStoS,thenf − 1isanisometryfromStoS(referto§6.5↓formoredetails).AninjectivemappingfromasurfaceSontoasurfaceSisanisometryiffthecoefficientsofthefirstfundamentalform(see§3.5↓)foranypatchonSareidenticaltothecorrespondingcoefficientsofthefirstfundamentalformofitsimageonS,thatis:(160)E = ẼF = FG = GwheretheunbarredandbarredE, F, Garethecoefficientsofthefirstfundamentalformofthetwosurfacesattheircorrespondingpoints.Asseenand

willbeseen,thecoefficientsofthefirstfundamentalformarethesameasthecoefficientsofthesurfacecovariantmetrictensor,thatis:(161)a11 = Ea12 = a21 = Fa22 = Gandhencetheseconditionsmeanthatthetwosurfaceshavethesamemetricattheircorrespondingpoints.Themappingthatpreservesdistancesbutitisnotinjectiveisdescribedaslocalisometry.Thestatementabouttheequalityofthecoefficientsofthefirstfundamentalformonthetwosurfacesalsoappliestolocalisometry.Sinceintrinsicpropertiesaredependentonlyonthecoefficientsofthefirstfundamentalformofthesurface,intrinsicpropertiesofthesurfaceareinvariantwithrespecttoisometricmappings.Asaconsequenceoftheequalityofcorrespondinglengthsoftwoisometricsurfaces,thecorrespondinganglesarealsoequal.However,thereverseisnottrue,thatisamappingthatattainstheequalityofcorrespondinganglesdoesnotnecessarilyensuetheequalityofcorrespondinglengths.Hence,isometricmappingismorerestrictivethanconformalmapping.Thismeansthateveryisometricmappingisconformalbutnoteveryconformalmappingisisometric,soisometricmappingisasubsetofconformalmapping.ThiscanbeseenbycomparingEqs.159↑and160↑wherethelattercorrespondstotheformerwithq = 1.Infact,conformalmappingcanbesetupbetweenanytwosurfacesandinmanydifferentwaysbutthisisnotalwayspossibleforisometricmapping.Isometricmappingalsopreservesareasofmappedsurfacessinceitpreserveslengthsandangles.Asurfacegeneratedbythecollectionofallthetangentlinestoagivenspacecurveiscalledthe“tangentsurface”ofthecurvewhilethetangentlinesarecalledthegeneratorsortherulingsofthesurface.Similarly,a“branch”ofthetangentsurfaceofacurveCatagivenpointPonthecurvereferstothetangentlineofCatP.Inthiscontext,weshouldremarkthatthe“tangentsurface”ofacurveshouldnotbeconfusedwiththeaforementioned“tangentplane”ofasurfaceatagivenpoint.Thetangentsurfaceofacurvemaybedemonstratedvisuallybyatautflexiblestringconnectedtothecurvewhereitscansthesurfacewhilebeingdirectedtangentiallyateachpointofthecurveatitsbase.However,thetautstringvisualizationshouldextendtobothtangentialdirectionstogivethefullextentofthetangentsurface(see§6.6↓formoredetails).IfCeisaspacecurvewithatangentsurfaceSTandCiisacurveembeddedinSTandisorthogonaltoallthetangentlinesofCeattheirintersectionpoints,thenCiiscalledaninvoluteofCewhileCeiscalledanevoluteofCi(see§5.3↓).

3.2 MathematicalDescriptionofSurfacesWestartbyassumingaparametricrepresentationforthesurface,whereeachoneofthespacecoordinates(x, y, z)onthesurfaceisarealdifferentiablefunctionofthetwosurfacecoordinates(u, v).ThepositionvectorofapointPonthesurfaceasafunctionofthesurfacecoordinatesisthengivenby:(162)r(u, v) = x(u, v)e1 + y(u, v)e2 + z(u, v)e3wheretheindexedearetheCartesianorthonormalbasisvectorsinthethreedirections.Itisalsoassumedthat∂urand∂vrarelinearlyindependentandhencetheyarenotparalleloranti-parallel,thatis:(163)(∂r ⁄ ∂u) × (∂r ⁄ ∂v) ≠ 0Asseenbefore,thisisasufficientandnecessaryconditionforthesurfacetobe“regular”atagivenpoint.Thepointisalsodescribedas“regular”;otherwiseitis“singular”iftheconditionisviolated.ThesurfaceisregularonΩ,aclosedsubsetofℝ2,ifitisregularateachinteriorpointofΩ.Theregularityconditionguaranteesthatthesurfacemappingisone-to-oneandpossessesacontinuousinverse.Italsoensurestheexistenceofatangentplaneandanormalvectorwherethisconditionissatisfied.ToexpressthepositionvectorofPintensornotation,were-labelthespaceandsurfacecoordinatesas:(164)(x, y, z) ≡ (x1, x2, x3)(u, v) ≡ (u1, u2)andhencethepositionvectorofEq.162↑becomes:(165)r(u1, u2) = xi(u1, u2) eiwherei = 1, 2, 3.Wenotethatrelabelingthesurfacecoordinatesisnotnecessaryinthisnotationbutitwillbeusefulinthefutureforothertensornotations.Todefineasurfacegridservingasacurvilinearpositioningsystemforthesurface,oneofthesurfacecoordinatevariablesisheldfixedinturnwhiletheotherisvaried(Figs.20↑and30↓).Hence,eachoneofthefollowingtwosurfacefunctions:(166)r(u1, c2)r(c1, u2)definesacoordinatecurveforthesurface,wherec1andc2aregivenrealconstants.Thesetwocoordinatecurvesmeetatthecommonsurfacepoint(c1, c2).Thegridisthengeneratedbyvaryingc1andc2uniformlytoobtaincoordinatecurvesatregularintervalsinitsdomain.

Figure30 Surfacecoordinategridwiththesurfacecovariantbasisvectors,E1andE2,andthenormalvectortothesurfacenataparticularpointonthesurfacewhere“CC”standsfor“coordinatecurves”.Correspondingtoeachoneofthesurfacecoordinatecurvesintheaboveorder,atangentvectortothecurveatagivenpointonthecurveisdefinedby:(167)Eα = ∂r ⁄ ∂uα = (∂xi ⁄ ∂uα)ei = xiαeiwherethederivativesareevaluatedatthatpoint,andα = 1, 2andi = 1, 2, 3.Soinbrief,E1istangenttother(u1, c2)coordinatecurveandE2istangenttother(c1, u2)coordinatecurve.Thesetangentvectorsserveasasetofbasisvectorsforthesurface,andforeachregularpointonthesurfacetheygenerate,bytheirlinearcombination,anyvectorinthesurfaceatthatpoint(seeFootnote11in§8↓).Theyalsodefine,bytheirlinearcombination,aplanetangenttothesurfaceatthatpoint.TheplanegeneratedbythelinearcombinationofE1andE2istheaforementionedtangentspace,TPS,tothesurfaceatpointPasdescribedearlier,andhenceE1(u1P, u2P)andE2(u1P, u2P)formabasisforthisspacewherethesubscriptPisareferencetothepointP.Weshouldremarkthatthesurfacebasisvectors,E1andE2,aredefinedonallpointsofthesurfaceandnotonlyonthepointsoftheabove-describedregularlyspacedgridofcoordinatecurveswhichispresentedinthatwayforpedagogicalreasons.Also,acoordinatecurveisanysurfacecurvealongwhichonlyonecoordinatevariable,u1oru2,variesregardlessofbeingpartoftheabovegridornot.Anotherimportantremarkisthatthesurfacecoordinatecurvesareorthogonalateverypointonthesurfaceiffthesurfacemetrictensor(see§3.3↓)isdiagonaleverywhereonthesurface.ThisisequivalenttohavinganidenticallyvanishingF,whichisthecoefficientofthefirstfundamentalform,ascanbeseenforexamplefromEq.235↓where

thedotproductwillvanishduetotheorthogonalityofE1andE2whicharethetangentstothecoordinatecurves.Whenthisconditionissatisfied,thecoordinatesystemofthesurfaceisdescribedasorthogonal.Also,thesurfacecoordinatecurvesareorthogonalatanyparticularpointonthesurfaceiffF = 0atthatpointevenifthisconditionisnotsatisfiedovertheentiresurface.Followingthedefinitionofthesurfacebasisvectors,E1andE2,anormalvectortothesurfaceatthegivenpointisthendefinedasthecrossproductofthesetangentbasisvectors:E1 ×  E2.Thisnormalvector,likeE1andE2,isafunctionofpositiononthesurfaceandhenceingeneralitvariescontinuously,inmagnitudeanddirection,asitmovesaroundthesurface.Thisnormalvectorcanbescaledbyitsmagnitudetoproduceaunitnormalvectorntothesurfaceatthatpoint,thatis(seeFootnote12in§8↓):(168)n = (E1 ×  E2) ⁄ | E1 ×  E2| = (E1 ×  E2) ⁄ √(a)whereaisthedeterminantofthesurfacecovariantmetrictensor(see§3.3↓).Basedonthecrossproductrule,thevectorsofthetriadE1,  E2,  nformarighthandedsystem(seeFig.30↑).OndotproductingbothsidesofEq.168↑withn,whichisaunitvector,weobtain:(169)n⋅(E1 ×  E2) = √(a)WemayalsotakethemodulusofbothsidesofEq.168↑(orjustcomparethedenominatorsofthesecondequalityofEq.168↑)toobtain:(170)|E1 ×  E2| = √(a)Thesurfacebasisvectors,Eα,aresymbolizedinfulltensornotationby:(171)[Eα]i ≡ Ei

α = ∂xi ⁄ ∂uα = xiα(i = 1, 2, 3andα = 1, 2)andhencetheycanberegardedas3Dcontravariantspacevectorsoras2Dcovariantsurfacevectors(referto§3.3↓forfurtherdetails).Itcanbeshownthatthecovariantformoftheunitnormalvectorntothesurfaceisgiveninfulltensornotationby:(172)ni = (1 ⁄ 2)εαβεijkxjαxkβwherexjα = (∂xj ⁄ ∂uα)andsimilarlyforxkβ,andεαβandεijkaretheabsolutepermutationtensorsforthesurfaceandspace.Theimplicationofthisequation,whichdefinesnintermsofthesurfacebasisvectorsxjαandxkβ,isthatnisaspacevectorwhichisindependentofthechoiceofthesurfacecoordinates,u1andu2,insupportofthegeometricintuition.Sincenisnormaltothesurface,wehave:(173)gijnixjα = 0whichisastatement,intensornotation,thatnisorthogonaltoeveryvectorinthetangentspaceofthesurfaceatthegivenpoint.Inthisequation,gijisthespacemetrictensor.AlthoughE1andE2arelinearlyindependenttheyarenotnecessarilyorthogonalorofunitlength.However,theycanbeorthonormalizedasfollows:(174)Ê1 =  E1 ⁄ | E1| =  E1 ⁄ √(a11)Ê2 = (a11 E2 − a12 E1) ⁄ √(a11a)whereaisthedeterminantofthesurfacecovariantmetrictensor(see§3.3↓),theindexedaarethecoefficientsofthistensor,andthehattedvectorsareorthonormalbasisvectors,thatis:(175)Ê1⋅ Ê1 = 1Ê2⋅ Ê2 = 1Ê1⋅ Ê2 = 0ThiscanbeverifiedbyconductingthedotproductsofthelastequationusingthevectorsdefinedinEq.174↑.Thetransformationrulesfromonesurfacecoordinatesystemtoanothersurfacecoordinatesystem,notatedwithunbarred(u1, u2)andbarred(ũ1, ũ2)symbolsrespectively,where:(176)u1 = u1(ũ1, ũ2)u2 = u2(ũ1, ũ2)(177)ũ1 = ũ1(u1, u2)ũ2 = ũ2(u1, u2)aresimilartothegeneralrulesforthetransformationbetweencoordinatesystemsinageneralnDspace,asexplainedinthetextbooksoftensoranalysis.Followingatransformationfromanunbarredsurfacecoordinatesystemtoabarredsurfacecoordinatesystem,thesurfacebecomesafunctionofthebarredcoordinates,andanewsetofbasisvectorsforthesurface,whicharethetangentstothecoordinatecurvesofthebarredsystem,aredefinedbythefollowingequations:(178)Ẽ1 = ∂r ⁄ ∂ũ1 = (∂r ⁄ ∂u1)(∂u1 ⁄ ∂ũ1) + (∂ r ⁄ ∂u2)(∂u2 ⁄ ∂ũ1) = E1(∂u1 ⁄ ∂ũ1) +  E2(∂u2 ⁄ ∂ũ1)(179)Ẽ2 = ∂r ⁄ ∂ũ2 = (∂r ⁄ ∂u1)(∂u1 ⁄ ∂ũ2) + (∂ r ⁄ ∂u2)(∂u2 ⁄ ∂ũ2) = E1(∂u1 ⁄ ∂ũ2) +  E2(∂u2 ⁄ ∂ũ2)Theseequations,whichcorrelatethesurfacebasisvectorsinthebarredandunbarredsurfacecoordinatesystems,canbecompactlypresentedintensornotationas:(180)∂xi ⁄ ∂ũα = (∂xi ⁄ ∂uβ)(∂uβ ⁄ ∂ũα)wherei = 1, 2, 3andα, β = 1, 2.Asetofcontravariantbasisvectorsforthesurfacemayalsobedefinedasthegradientofthesurfacecoordinatecurves,thatis:(181)Eα = ∇uαIntensornotation,thisbasismaybegivenby:(182)[Eα]i ≡ Eα

i = ∂uα ⁄ ∂xi = xαiHence,thesebasisvectorscanberegardedas2Dcontravariantsurfacevectorsoras3Dcovariantspacevectors.Thecontravariantandcovariantformsofthesurfacebasisvectors,EαandEα,areobtainedfromeachotherbytheindexshiftingoperatorforthesurface,thatis:(183)Eα = aαβEβ

Eα = aαβEβwheretheindexedaarethecovariantandcontravariantformsofthesurfacemetrictensor(see§3.3↓).Thecontravariantandcovariantformsofthesurfacebasisvectors,EαandEα,arereciprocalsystemsandhencetheysatisfythefollowingreciprocityrelations:(184)Eα⋅Eβ = δαβ ≡ aαβ

Eα⋅Eβ = δαβ ≡ aαβwheretheindexedδandaarethemixedtypeoftheKroneckerdeltaandsurfacemetrictensors.Thesurfacebasisvectorsintheircovariantandcontravariantforms,xiαandxβj,andtheunitnormalvectortothesurfacenkarelinkedbythefollowingrelation:(185)xiα = εijkεαβxβjnkThisequationmeansthatthegivenproduct(whichlookslikeavectorcrossproduct)ofthesurfacecontravariantbasisvectorxβjandtheunitnormalvectornkproducesasurfacecovariantbasisvectorxiαandhenceitisperpendiculartoboth.Beingasurfacebasisvectorimpliesorthogonalitytotheunitnormalvector,whilebeingacovariantsurfacebasisvectorimplieshereorthogonalitytothecontravariantsurfacebasisvector.

3.3 SurfaceMetricTensorThesurfacemetrictensorisanabsolute,rank-2,2 × 2tensor.Aswewillsee,allformsofthistensor(i.e.covariant,contravariantandmixed)aresymmetric.Indifferentialgeometry,thesurfacemetrictensoraαβmaybecalledthefirstgroundformorthefundamentalsurfacetensor.Weremarkthatthecoefficientsofthemetrictensorarerealnumbers.FollowingtheexampleofthemetricingeneralnDspaces,thesurfacemetrictensorofa2Dsurfaceembeddedina3DEuclideanflatspacewithmetricgij = δijisgiveninitscovariantformby:(186)aαβ = Eα⋅Eβ = (∂r ⁄ ∂uα)⋅(∂r ⁄ ∂uβ) = (∂xi ⁄ ∂uα)(∂xi ⁄ ∂uβ)wheretheindexedxanduarethespaceCartesiancoordinatesandthesurfacecoordinatesrespectively,andi = 1, 2, 3andα, β = 1, 2.ThesurfaceandspacemetrictensorsinageneralRiemannianspacewithgeneralmetricgijarerelatedby:(187)aαβ = Eα⋅Eβ = (∂r ⁄ ∂uα)⋅(∂r ⁄ ∂uβ) = gij(∂xi ⁄ ∂uα)(∂xj ⁄ ∂uβ) = gijxiαxjβwhereaαβandgijarerespectivelythesurfaceandspacecovariantmetrictensors,theindexedxanduaregeneralcoordinatesofthespaceandsurfacerespectively,andi, j = 1, 2, 3andα, β = 1, 2.ItisobviousthatEq.186↑isaspecialinstanceofEq.187↑forthecaseofaflatspacewithaCartesiansystemwherethespacemetricistheunitytensor.Eq.187↑isthefundamentalrelationthatprovidesthecruciallinkbetweenthesurfaceanditsenvelopingspace.Asindicatedbefore,thepartialderivativesinthisrelation,(∂xi ⁄ ∂uα)and(∂xj ⁄ ∂uβ),maybeconsideredascontravariantrank-13Dspacetensorsorascovariantrank-12Dsurfacetensors.Hence,atensorlike(∂xi ⁄ ∂uα)isusuallylabeledasxiαtoindicatethatitrepresentstwosurfacevectorswhicharecontravariantly-transformedwithrespecttothethreespacecoordinatesxi:(188)xi1 = (∂x1 ⁄ ∂u1, ∂x2 ⁄ ∂u1, ∂x3 ⁄ ∂u1)xi2 = (∂x1 ⁄ ∂u2, ∂x2 ⁄ ∂u2, ∂x3 ⁄ ∂u2)orthreespacevectorswhicharecovariantly-transformedwithrespecttothetwosurfacecoordinatesuα:(189)x1α = (∂x1 ⁄ ∂u1, ∂x1 ⁄ ∂u2)x2α = (∂x2 ⁄ ∂u1, ∂x2 ⁄ ∂u2)x3α = (∂x3 ⁄ ∂u1, ∂x3 ⁄ ∂u2)AnysurfacevectorAα(α = 1, 2),definedasalinearcombinationofthesurfacebasisvectorsE1andE2,canalsobeconsideredasaspacevectorAi(i = 1, 2, 3)wherethetworepresentationsarelinkedthroughtherelation:(190)Ai = (∂xi ⁄ ∂uα)Aα = xiαAαwherei = 1, 2, 3andα = 1, 2.Now,sincewehave:(191)aαβAαAβ = gijxiαxjβAαAβ(Eq.187↑) = gijxiαAαxjβAβ

 = gijAiAj(Eq.190↑)thenweseethatthetworepresentationsareequivalent,thatistheydefineavectorofthesamemagnitudeanddirection.Thecontravariantformofthesurfacemetrictensorisdefinedastheinverseofthesurfacecovariantmetrictensor,thatis:(192)aαγaγβ = δαβaαγaγβ = δαβSincethefirstfundamentalformispositivedefinite(see§3.5↓),andhencea > 0,theexistenceofaninverseisguaranteed.SimilartothemetrictensoringeneralnDspaces,thecovariantandcontravariantformsofthesurfacemetrictensor,aαβandaαβ,areusedforloweringandraisingindicesandrelatedtensoroperations.Thecovarianttypeofthesurfacemetrictensoraαβisgiveninmatrixformby:(193)

wherethecoefficientsaαβareasdefinedabove(seeEq.187↑),andE, F, Garethecoefficientsofthefirstfundamentalform(referto§3.5↓).Becausethecontravariantformofthesurfacemetrictensoraαβistheinverseofitscovariantform,itisgivenby:(194)

wherethesymbolsareasdefinedpreviously.Asseenintheaboveequation,thistensorissymmetricandhencea12 = a21.Asindicatedbefore,themixedformofthesurfacemetrictensoraαβistheidentitytensor,thatis:(195)

Followingthestyleofthespacemetrictensor,thesurfacemetrictensortransformsbetweenthebarredandunbarredsurfacecoordinatesystemsas:(196)ãαβ = aγδ(∂uγ ⁄ ∂ũα)(∂uδ ⁄ ∂ũβ)wheretheindexedãandaarethesurfacecovariantmetrictensorsinthebarredandunbarredsystemsrespectively.Thecontravariantandmixedformsofthesurfacemetrictensoralsofollowsimilartransformationrulestotheircounterpartsofthespacemetrictensor.Similartothedeterminantsofthespacemetric,thedeterminantsofthesurfacemetricinthebarredandunbarredcoordinatesystemsarelinkedthroughtheJacobianoftransformationbythefollowingrelation:(197)ã = J2awhereãandaarethedeterminantsofthecovariantformofthesurfacemetrictensorinthebarredandunbarredsystemsrespectivelyandJ( = |∂u ⁄ ∂ũ|)istheJacobianofthetransformationbetweenthetwosurfacesystems(referto§1.4.3↑).ThisrelationcanbeobtaineddirectlyfromEq.196↑bytakingthedeterminantofthetwosidesofthatequation.TheChristoffelsymbolsofthefirstkind[αβ, γ]forthesurfacearelinkedtothesurfacebasisvectorsandtheirpartialderivativesbythefollowingrelation:(198)[αβ, γ] = (∂Eα ⁄ ∂uβ)⋅EγHence,therelationbetweenthepartialderivativeofthesurfacecovariantmetrictensorandtheChristoffelsymbolsofthefirstkindisgivenby:(199)∂aαβ ⁄ ∂uγ = ∂(Eα⋅Eβ) ⁄ ∂uγ = (∂Eα ⁄ ∂uγ)⋅Eβ + Eα⋅(∂Eβ ⁄ ∂uγ) = [αγ, β] + [βγ, α]AsimilarrelationbetweenthederivativeofthesurfacemetrictensorandtheChristoffelsymbolsofthesecondkindcanbeobtainedfromtheprevious

equationbyusingtheindexshiftingoperatorofthesurface:(200)∂aαβ ⁄ ∂uγ = aδβΓδαγ + aδαΓδβγForaMongepatchoftheformr(u, v) = (u, v, f(u, v)),thesurfacecovariantmetrictensoraisgivenby:(201)

wherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates,andISisthetensorofthefirstfundamentalformofthesurface(see§3.5↓).Itisnoteworthythatscalingasurfaceupordownbyaconstantfactorc > 0,whichisequivalenttoscalingallthedistancesonthesurfacebythatfactor,canbedonebymultiplyingthesurfacemetrictensorbyc2.Thiscanbeseen,forexample,fromEq.187↑.Inthefollowingsubsections,weinvestigatearclength,areaandanglebetweentwocurvesonasurface.Alltheseentitiesdependintheirdefinitionandquantificationonthemetrictensor.Wewillseethatalltheseentities,duetotheirexclusivedependenceonthemetrictensor,areinvariantunderisometrictransformations.WewillalsoseethattheirgeometricandtensorformulationsareidenticaltothoseinageneralnDspacewiththeuseofthesurfacemetrictensorandthesurfacerepresentationoftheinvolvedquantities.

3.3.1 ArcLengthThelengthofaninfinitesimallineelementofasurfacecurverepresentstheresultantgrowthalongtheelementintheuandvdirectionsbetweenitstwoendpoints(seeFig.31↓).FollowingtheexampleofthelengthofanelementofarcofacurveembeddedinageneralnDspace,thelengthofanelementofarcofacurveona2Dsurface,ds,isgiveninitsgeneralformby(seeEqs.167↑and187↑):(202)(ds)2 = dr⋅dr = (∂r ⁄ ∂uα)⋅(∂r ⁄ ∂uβ)duαduβ = Eα⋅Eβduαduβ = aαβduαduβwhereaαβisthecovarianttypeofthesurfacemetrictensor,risthespatialrepresentationofthesurfacecurveandα, β = 1, 2.

Figure31 Thelengthofaninfinitesimallineelement,ds,ofasurfacecurveCwhereduanddvareusedtolabelinfinitesimalelementgrowthsonthecoordinatecurves(CC).Fromtheaboveformulawehavethefollowingidentitywhichisvalidateachpointofanaturallyparameterizedsurfacecurve:(203)aαβ(duα ⁄ ds)(duβ ⁄ ds) = 1Basedontheaboveformula(Eq.202↑),thelengthofasegmentofat-parameterizedcurvebetweenastartingpointcorrespondingtot = t1andaterminalpointcorrespondingtot = t2isgivenby:(204)

whereI ⊂ ℝisanintervalonthereallineandE, F, Garethecoefficientsofthefirstfundamentalform.Thefourthequalityisbasedontheequality:a12 = a21becausethemetrictensorissymmetric,asseenearlier.ForaMongepatchoftheformr(u, v) = (u, v, f(u, v)),thelengthofanelementofarcofasurfacecurveisgivenby:(205)ds = √([1 + fu2]dudu + 2fufvdudv + [1 + fv2]dvdv)wherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates.ThelastequationcanbeobtaineddirectlyfromEq.202↑usingthecoefficientsofthemetrictensorfromEq.201↑.Itisnoteworthythatthelengthofasurfacecurveisanintrinsicpropertysinceitdependsonthemetrictensoronly.Also,thelengthofasurfacecurveisinvariantwithrespecttothetypeofparameterizationandisometrictransformations.

3.3.2 SurfaceAreaTheareaofaninfinitesimalsurfaceelementofaspacesurfacerepresentstheresultantgrowthintheelementasaresultofthegrowthintheuandvdirectionsalongtheboundarycurvesthatdefinetheelement(seeFig.32↓).

Figure32 Theareaofaninfinitesimalsurfaceelement,dσ,ofaspacesurfacewhereduanddvareusedtolabelinfinitesimalelementgrowthsonthecoordinatecurvesand“CC”standsfor“coordinatecurve”.Theareaofaninfinitesimalelementofasurface,dσ,intheneighborhoodofapointPonthesurfaceisgivenby(seeEqs.170↑and187↑):(206)dσ = |dr1 × dr2| = |E1 ×  E2|du1du2 = √(|E1|2| E2|2 − [ E1⋅ E2]2)du1du2 = √(a11a22 − [a12]2)du1du2 = √(EG − F2)du1du2 = √(a)du1du2whereE1andE2arethesurfacecovariantbasisvectors,E, F, Garethecoefficientsofthefirstfundamentalform,a( = a11a22 − a12a21)isthedeterminantofthesurfacecovariantmetrictensorandtheindexedaareitselements.AllthequantitiesintheseexpressionsbelongtothepointP.Wealsoassumethatdu1du2ispositive.TheareaofasurfacepatchS:Ω → ℝ3,whereΩisapropersubsetoftheℝ2parametersplane,isgivenby:(207)σ = ∬Ωdσ = ∬Ω√(a11a22 − [a12]2)du1du2 = ∬Ω√(EG − F2)du1du2 = ∬Ω√(a)du1du2wherethesymbolsareasdefinedabove.ThepatchSshouldbeinjective,sufficientlydifferentiable,andregularontheinteriorofΩ.Theaboveformulaefortheareaarereminderofthevolumeformulaeina3Dspace,sotheareacanberegradedasavolumeina2Dspace.Asstatedbefore,theareasoftwocorrespondingsurfaceelementsandsurfacepatchesontwoisometricsurfacesareequal.Thiscanbeseenfromtheaboveformulaesincethemetrictensorisidenticalatthecorrespondingpointsoftwoisometricsurfaces.ForaMongepatchoftheformr(u, v) = (u, v, f(u, v)),thesurfaceareaisgivenby:(208)σ = ∬Ω√(1 + fu2 + fv2)dudvwherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates.ThelastequationcanbeobtaineddirectlyfromEq.207↑usingthecoefficientsofthemetrictensorfromEq.201↑.

3.3.3 AngleBetweenTwoSurfaceCurvesTheanglebetweentwosufficientlysmoothsurfacecurvesintersectingatagivenpointonthesurfaceisdefinedastheanglebetweentheirtangentvectorsatthatpoint(Fig.33↓).Astherearetwooppositedirectionsforeachcurve,correspondingtothetwosensesoftraversingthecurve,therearetwomainanglesθ1andθ2suchthatθ1 + θ2 = π.Theprincipalanglebetweenthetwocurvesisusuallytakenasthesmallerofthetwoanglesandhencethedirectionsaredeterminedaccordingly.Infact,therearestilltwopossibilitiesforthedirectionsbutthishasnosignificanceasfarastheanglebetweenthetwocurvesisconcerned.

Figure33 Theanglebetweentwosurfacecurves,C1andC2,astheangleθbetweentheirtangents,t1andt2,atthepointofintersectionP.TheangleθbetweentwosurfacecurvespassingthroughagivenpointPonthesurfacewithtangentvectorsAandBatPisgivenby:(209)cosθ = (A⋅B) ⁄ (|A||B|) = (aαβAαBβ) ⁄ [√(aγδAγAδ)√(aϵζBϵBζ)]wheretheindexedaaretheelementsofthesurfacecovariantmetrictensorandtheGreekindicesrunover1, 2.IfAandBaretwounitsurfacevectorswithsurfacerepresentationsAαandBβandspacerepresentationsAiandBjthentheangleθbetweenthetwovectorsisgivenby:(210)cosθ = aαβAαBβ(Eq.209↑) = gijxiαxjβAαBβ(Eq.187↑) = gijAiBj(Eq.190↑)whereα, β = 1, 2andi, j = 1, 2, 3.Hence,thesurfaceandspacerepresentationsofthetwovectorsdefinethesameangle.ThevectorsAandB(whetherunitvectorsornot)areorthogonaliffaαβAαBβ = 0atP.Asseenearlier,thecoordinatecurvesatagivenpointPonasurfaceareorthogonaliffa12 ≡ F = 0atP.ThiscanbeseenfromEqs.187↑and235↓wherethedotproductswillvanishduetotheorthogonalityofE1andE2whicharethetangentstothecoordinatecurves.Thecorrespondinganglesoftwoisometricsurfaces,likethecorrespondinglengthsandareas,areequalascanbeseenfromtheaboveformulaeconsideringthatthemetrictensorisidenticalatthecorrespondingpoints.However,thereverseisnottrueingeneral,thatistheequalityofanglesontwosurfacesrelatedbyagivenmapping,asintheconformalmapping,doesnotleadtotheequalityofthecorrespondinglengthsonthetwomappedsurfaces,asexplainedbefore.Thisisduetothefactthattheequalityofanglesrequirestheproportionalityofthetwometrictensorsatthecorrespondingpointsbutnotnecessarilytheequality(seeEq.159↑).ThismaybeconcludedfromEq.209↑whereanyproportionalityfactorwillbecanceledout.Thesineoftheangleθbetweentwosurfaceunitvectors,AandB,isgivenby:(211)sinθ = εαβAαBβwhereεαβisthe2Dabsolutepermutationtensor.ThesineinthisformulaisnumericallyequalinmagnitudetotheareaoftheparallelogramwithsidesAandB.Basedonthelastformula,thesufficientandnecessaryconditionforAandBtobeorthogonalisthat:(212)|εαβAαBβ| = 1

3.4 SurfaceCurvatureTensorThesurfacecurvaturetensorisanabsolute,rank-2,2 × 2tensor.Aswewillsee,thecovariantandcontravariantformsofthistensoraresymmetric.Thesurfacecurvaturetensorbαβmayalsobecalledthesecondgroundform.Wenotethatthecoefficientsofthecurvaturetensorarerealnumbers.Theelementsofthesurfacecovariantcurvaturetensor,bαβ,aregivenby:(213)bαβ =  − (∂r ⁄ ∂uα)⋅(∂n ⁄ ∂uβ) =  − Eα⋅(∂n ⁄ ∂uβ) =  − [(∂r ⁄ ∂uα)⋅(∂n ⁄ ∂uβ) + (∂r ⁄ ∂uβ)⋅(∂n ⁄ ∂uα)] ⁄ 2andalsoby:(214)bαβ = (∂2 r ⁄ ∂uα∂uβ)⋅n = (∂Eα ⁄ ∂uβ)⋅n = [∂Eα ⁄ ∂uβ]⋅[(E1 ×  E2) ⁄ √(a)] = [(∂Eα ⁄ ∂uβ)⋅(E1 ×  E2)] ⁄ √(a)whereEq.168↑isusedinthelasttwosteps.Wenotethattheequality:(215)(∂Eα ⁄ ∂uβ)⋅n =  − Eα⋅(∂n ⁄ ∂uβ)whichisseenbycomparingEqs.213↑and214↑isbasedontheequality:(216)∂(Eα⋅n) ⁄ ∂uβ = ∂(0) ⁄ ∂uβ = 0plustheproductruleofdifferentiation.ConsideringEq.214↑,thesymmetryofthesurfacecovariantcurvaturetensor(i.e.b12 = b21)followsfromthefactthat:(217)∂Eα ⁄ ∂uβ = ∂2 r ⁄ ∂uβ∂uα = ∂2 r ⁄ ∂uα∂uβ = ∂Eβ ⁄ ∂uαInfulltensornotation,thesurfacecovariantcurvaturetensorisgivenby:(218)bαβ = (εγδεijkxiα;βxjγxkδ) ⁄ 2 = (εijkxiα;βxj1xk2) ⁄ √(a)whereaisthedeterminantofthesurfacecovariantmetrictensorandtheindexedε’saretheabsolutepermutationtensorsof2Dand3Dspacesintheircontravariantandcovariantforms.ThisformulawillsimplifytothefollowingwhenthespacecoordinatesarerectangularCartesian:(219)bαβ = [ϵijk(∂2xi ⁄ ∂uα∂uβ)xj1xk2] ⁄ √(a)Thesurfacecurvaturetensorobeysthesametransformationrulesasthesurfacemetrictensor.Forexample,forthetransformationbetweenthebarredandunbarredsurfacecoordinatesystemswehave(seeEq.196↑):(220)bαβ = bγδ(∂uγ ⁄ ∂ũα)(∂uδ ⁄ ∂ũβ)wheretheindexedbandbarethecoefficientsofthecovariantcurvaturetensorinthebarredandunbarredsystemsrespectively.Similarly,wehave(seeEq.197↑):(221)b = J2bwherebandbarethedeterminantsofthesurfacecovariantcurvaturetensorinthebarredandunbarredsystemsrespectivelyandJ( = |∂u ⁄ ∂ũ|)istheJacobianofthetransformationbetweenthetwosurfacecoordinatesystems.Thesurfacecovariantcurvaturetensorisgiveninmatrixformby:(222)

wheree, f, garethecoefficientsofthesecondfundamentalformofthesurface(referto§3.6↓).Thisisasymmetricmatrixasindicatedearlierandasseeninthelastpartoftheequation.Themixedformofthesurfacecurvaturetensorbαβisgivenby:(223)

whereE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalformsanda = EG − F2isthedeterminantofthesurfacecovariantmetrictensor.Asseen,thecoefficientsofbαβdependonthecoefficientsofboththefirstandsecondfundamentalforms.Themixedformbαβ = bαγaγβisthetransposeoftheaboveform.Thecontravariantformofthesurfacecurvaturetensorisgivenby:(224)

where

A = eG2 − 2fFG + gF2

B = fEG − eFG − gEF + fF2

C = fEG − eFG − gEF + fF2

D = gE2 − 2fEF + eF2.Likethecovariantform,thecontravariantformisasymmetrictensorasindicatedbeforeandasseenabove.Aswewillsee,thetraceofthesurfacemixedcurvaturetensorbαβistwicethemeancurvatureH(see§4.6↓),whileitsdeterminantistheGaussiancurvatureK(see§4.5↓),thatis:(225)H = tr(bαβ) ⁄ 2K = det(bαβ)Becausethetraceandthedeterminantofatensorareitsmaintwoinvariantsunderpermissiblecoordinatetransformations(refertosimilaritytransformationsoflinearalgebra),thenHandKareinvariant,aswillbeestablishedlaterinthebook.ThesurfacecovariantcurvaturetensorofaMongepatchoftheformr(u, v) = (u, v, f(u, v))isgivenby:(226)

wherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates.Sincefuv = fvu,thetensorissymmetricasitshouldbe.Theequality:fuv = fvuisbasedonthewellknowncontinuityconditionwhichisfullyexplainedinanystandardtextbookoncalculus.ThecovariantcurvaturetensorofasurfaceandtheRiemann-Christoffelcurvaturetensorofthefirstkindarelinkedbythefollowingrelation:(227)Rαβγδ = bαγbβδ − bαδbβγThisrelationmayalsobegivenintermsoftheRiemann-Christoffelcurvaturetensorofthesecondkindusingtheindexraisingoperatorforthesurface,thatis:(228)aαωRωβγδ = Rα

βγδ = bαγbβδ − bαδbβγFromEqs.88↑and227↑,itcanbeconcludedthatthesurfacecovariantcurvaturetensorandthesurfaceChristoffelsymbolsofthefirstandsecondkindarerelatedby:(229)bαγbβδ − bαδbβγ = (∂[βδ, α] ⁄ ∂γ) − (∂[βγ, α] ⁄ ∂δ) + [αδ, ω]Γω

βγ − [αγ, ω]Γωβδ

Similarly,fromEqs.89↑and228↑,itcanbeconcludedthatthesurfacecurvaturetensorandthesurfaceChristoffelsymbolsofthesecondkindarerelatedby:(230)bαγbβδ − bαδbβγ = (∂Γα

βδ ⁄ ∂γ) − (∂Γαβγ ⁄ ∂δ) + Γω

βδΓαωγ − Γω

βγΓαωδ

Consideringthefactthatthe2DRiemann-Christoffelcurvaturetensorhasonlyonedegreeoffreedom(see§1.4.10↑)andhenceitpossessesasingleindependentnon-vanishingcomponentwhichisrepresentedbyR1212,weseethatEq.227↑hasonlyoneindependentcomponentwhichisgivenby:(231)R1212 = b11b22 − b12b21 = bwherebisthedeterminantofthesurfacecovariantcurvaturetensor.Equation227↑alsoshowsthateachoneofthefollowingprovisions:(232)Rαβγδ = 0andbαβ = 0(α, β, γ, δ = 1, 2)ifsatisfiedidenticallyisasufficientandnecessaryconditionforhavingaflat2Dspace.However,foraplanesurfaceallthecoefficientsoftheRiemann-Christoffelcurvaturetensorandthecoefficientsofthesurfacecurvaturetensorvanishidenticallythroughoutthesurface,butforasurfacewhichisisometrictoplanethecoefficientsoftheRiemann-Christoffelcurvaturetensorvanishidenticallybutnotnecessarilythecoefficientsofthesurfacecurvaturetensor.Thisisbasedonthefactthathavingazerodeterminantdoesnotimplyhavingazerotensor(seeEq.231↑).So,asstatedpreviously,theprovisionRαβγδ = 0isasufficientandnecessaryconditionforhavinganintrinsicallyflatspace,whiletheprovisionbαβ = 0isasufficientandnecessaryconditionforhavinganextrinsicallyflatspace.Havinganextrinsicallyflatspaceimplieshavinganintrinsicallyflatspacesinceacurvaturethatcannotbeobservedfromoutsidethespacecannotbeseenfrominsidethespace.FromEqs.88↑and227↑,weseethattheRiemann-ChristoffelcurvaturetensorcanbeexpressedintermsofthecoefficientsofthesurfacecurvaturetensoraswellasintermsofthecoefficientsofthesurfacemetrictensorwherethetwosetsofcoefficientsarelinkedthroughEq.229↑.ThisdoesnotmeanthatRiemann-Christoffelcurvatureisanextrinsicpropertybutitmeansthatsomeintrinsicpropertiescanalsobedefinedintermsofextrinsicparameters.Itshouldbenoticedthatthesignofthesurfacecurvaturetensor(i.e.thesignofitscoefficients)isdependentonthechoiceofthedirectionoftheunitnormalvectortothesurface,n,andhencethesignofthecoefficientswillbereversedifthedirectionofnisreversed.Thiscanbeseen,forexample,fromEq.214↑.

3.5 FirstFundamentalFormAsindicatedpreviously,thefirstfundamentalform,whichisbasedonthemetric,encompassesalltheintrinsicinformationaboutthesurfacethata2Dinhabitantofthesurfacecanobtainfrommeasurementsperformedonthesurfacewithoutappealingtoanexternaldimension.Intheoldbooks,thefirstfundamentalformmaybelabeledasthefirstfundamentalquadraticform.ThefirstfundamentalformISofthelengthofanelementofarcofacurveonasurfaceisaquadraticexpressiongivenby:(233)IS = (ds)2 = dr⋅dr = (∂r ⁄ ∂uα)⋅(∂r ⁄ ∂uβ)duαduβ = Eα⋅Eβduαduβ = aαβduαduβ = E(du1)2 + 2Fdu1du2 + G(du2)2whereE, F, G,whichingeneralarecontinuousvariablefunctionsofthesurfacecoordinatesu1andu2,aregivenby:(234)E = a11 = E1⋅ E1 = (∂r ⁄ ∂u1)⋅(∂ r ⁄ ∂u1) = gij(∂xi ⁄ ∂u1)(∂xj ⁄ ∂u1)(235)F = a12 = E1⋅ E2 = (∂r ⁄ ∂u1)⋅(∂ r ⁄ ∂u2) = gij(∂xi ⁄ ∂u1)(∂xj ⁄ ∂u2) = E2⋅ E1 = a21(236)G = a22 = E2⋅ E2 = (∂r ⁄ ∂u2)⋅(∂ r ⁄ ∂u2) = gij(∂xi ⁄ ∂u2)(∂xj ⁄ ∂u2)wheretheindexedaaretheelementsofthesurfacecovariantmetrictensor,theindexedxarethegeneralcoordinatesoftheenvelopingspaceandgijisitscovariantmetrictensor.ForaflatspacewithaCartesiancoordinatesystemxi,thespacemetricisgij = δijandhencetheaboveequationsbecome:(237)E = (∂xi ⁄ ∂u1)(∂xi ⁄ ∂u1)(238)F = (∂xi ⁄ ∂u1)(∂xi ⁄ ∂u2)(239)G = (∂xi ⁄ ∂u2)(∂xi ⁄ ∂u2)Thefirstfundamentalformcanbecastinthefollowingmatrixform:(240)

wherevisadirectionvector,vTisitstranspose,andISisthefirstfundamentalformtensorwhichisequaltothesurfacecovariantmetrictensor.Hence,thematrixassociatedwiththefirstfundamentalformisthecovariantmetrictensorofthesurface.WeremarkthatthenotationregardingthedotproductoperationbetweentwomatricesinthefirstlineofEq.240↑maynotbeastandardone.Wealsonotethatwhilethecoefficientsofthefirstfundamentalformdependonthepositiononthesurfaceandhencetheyarefunctionsofthesurfacecoordinatesbutnotthedirection,thefirstfundamentalformisafunctionofbothpositionanddirection.Thefirstfundamentalformisnotauniquecharacteristicofthesurfaceandhencetwogeometricallydifferentsurfacesasseenfromtheenvelopingspace,suchasplaneandcylinder,canhavethesamefirstfundamentalform.Suchsurfacesaredifferentextrinsicallyasseenfromtheembeddingspacealthoughtheyareidenticalintrinsicallyasviewedinternallyfromthesurfacebya2Dinhabitant.Thefirstfundamentalformispositivedefiniteatregularpointsof2Dsurfaces;henceitscoefficientsaresubjecttothefollowingconditions:(241)E > 0anddet(IS) = EG − F2 > 0Asindicatedearlier,theaboveconditionsimplyG > 0sincethesecoefficientsarereal.ItisnoteworthythattheconditionofpositivedefinitenessmaybeamendedtoallowformetricsbasedoncoordinatesystemswithimaginarycoordinatesasitisthecaseinthecoordinatesystemsoftheMinkowski

space(see§1.4.7↑),butthisisoutofthescopeofthepresentbookwhosefocusisthegeometryofspacecurvesandsurfacesinastaticsenseandhenceallmetricsconsideredinthisbookarepositivedefinite.Asindicatedabove,thefirstfundamentalformencompassestheintrinsicpropertiesofthesurfacegeometry.Hence,asseenin§3.3↑,thefirstfundamentalformisusedtodefineandquantifythingslikearclength,areaandanglebetweencurvesonasurfacebasedonitsqualificationasametric.Forexample,Eq.204↑showsthatthelengthofacurvesegmentonasurfaceisobtainedbyintegratingthesquarerootofthefirstfundamentalformofthesurfacealongthesegment.IfasurfaceS1canbemappedisometrically(see§3.1↑and§6.5↓)ontoanothersurfaceS2,thenthetwosurfaceshaveidenticalfirstfundamentalformsandidenticalfirstfundamentalformcoefficientsattheircorrespondingpoints,thatis(seeFootnote13in§8↓):(242)E1 = E2F1 = F2G1 = G2wherethesubscriptsarelabelsforthetwosurfaces.BasedonEq.201↑,foraMongepatchoftheformr(u, v) = (u, v, f(u, v)),thefirstfundamentalformisgivenby:(243)IS = (1 + fu2)dudu + 2fufvdudv + (1 + fv2)dvdvwherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates.

3.6 SecondFundamentalFormThemathematicalentitythatcharacterizestheextrinsicgeometryofasurfaceisthenormalvectortothesurface.Thisentitycanonlybeobservedexternallyfromoutsidethesurfacebyanobserverinareferenceframeinthespacethatenvelopsthesurface.Hence,thenormalvectorandallitssubsidiariesarestrangetoa2Dinhabitanttothesurfacewhocanonlyaccesstheintrinsicattributesofthesurfaceasrepresentedbyandcontainedinthefirstfundamentalform.Inbrief,the2Dinhabitanthasnonotionoftheextradimensionwhichembracesthenormalvector.Asaconsequence,thevariationofthenormalvectorasitmovesaroundthesurfacecanbeusedasanindicatortocharacterizethesurfaceshapefromanexternalpointofviewandthatishowthisindicatorisexploitedinthesecondfundamentalformtorepresenttheextrinsicgeometryofthesurfaceaswillbeseenfromtheforthcomingformulationsofthesecondfundamentalform.ThesecondfundamentalformIISofasurface,whichintheoldbooksmaybelabeledasthesecondfundamentalquadraticform,isdefinedbythefollowingquadraticexpression:(244)IIS =  − dr⋅dn =  − [(∂r ⁄ ∂uα)duα]⋅[(∂n ⁄ ∂uβ)duβ] =  − [(∂r ⁄ ∂u1)du1 + (∂ r ⁄ ∂u2)du2]⋅[(∂ n ⁄ ∂u1)du1 + (∂ n ⁄ ∂u2)du2] = e(du1)2 + 2fdu1du2 + g(du2)2wherethecoefficientsofthesecondfundamentalforme, f, garegivenby:(245)e =  − (∂r ⁄ ∂u1)⋅(∂ n ⁄ ∂u1) =  − E1⋅(∂ n ⁄ ∂u1)(246)f =  − [(∂r ⁄ ∂u1)⋅(∂ n ⁄ ∂u2) + (∂ r ⁄ ∂u2)⋅(∂ n ⁄ ∂u1)] ⁄ 2 =  − [E1⋅(∂ n ⁄ ∂u2) +  E2⋅(∂ n ⁄ ∂u1)] ⁄ 2(247)g =  − (∂r ⁄ ∂u2)⋅(∂ n ⁄ ∂u2) =  − E2⋅(∂ n ⁄ ∂u2)Intheaboveequations,r(u1, u2)denotesthespatialrepresentationofthesurface,n(u1, u2)istheunitnormalvectortothesurfaceandα, β = 1, 2.Thesecondfundamentalformmayalsobegivenby:(248)

whered2 risthesecondorderdifferentialofthepositionvectorrofanarbitrarypointonthesurfaceinthedirection(du1, du2).Asaresult,thecoefficientsofthesecondfundamentalformcanalsobegivenbythefollowingalternativeexpressions:(249)e = n⋅(∂E1 ⁄ ∂u1) =  − (∂n ⁄ ∂u1)⋅ E1(250)f = n⋅(∂E1 ⁄ ∂u2) = n⋅(∂E2 ⁄ ∂u1) =  − (∂n ⁄ ∂u2)⋅ E1 =  − (∂n ⁄ ∂u1)⋅ E2(251)g = n⋅(∂E2 ⁄ ∂u2) =  − (∂n ⁄ ∂u2)⋅ E2Thetwoformsofeachformulainthelastequationsarebasedonthefactthatn⋅Eα = 0,sincenisperpendiculartothesurfacebasisvectors,plustheproductruleofdifferentiation,thatis:(252)∂(n⋅Eα) ⁄ ∂uβ = ∂(0) ⁄ ∂uβ = n⋅(∂Eα ⁄ ∂uβ) + (∂n ⁄ ∂uβ)⋅Eα = 0 ⇒  n⋅(∂Eα ⁄ ∂uβ) =  − (∂n ⁄ ∂uβ)⋅Eα

Theequality:n⋅(∂2 E1) =  n⋅(∂1 E2)whichisseeninEq.250↑isbasedontheequality:∂αEβ = ∂βEαasstatedbefore(seeEq.217↑).FromEqs.249↑-251↑plusEq.168↑,wecanseethatthecoefficientsofthesecondfundamentalformmayalsobegivenbythefollowingexpressions:(253)e = [(E1 ×  E2)⋅(∂ E1 ⁄ ∂u1)] ⁄ √(a)(254)f = [(E1 ×  E2)⋅(∂ E1 ⁄ ∂u2)] ⁄ √(a)(255)g = [(E1 ×  E2)⋅(∂ E2 ⁄ ∂u2)] ⁄ √(a)wherea( = a11a22 − a12a21 = EG − F2)isthedeterminantofthesurfacecovariantmetrictensor.Likethecoefficientsofthefirstfundamentalform,thecoefficientsofthesecondfundamentalformare,ingeneral,continuousvariablefunctionsofthesurfacecoordinatesu1andu2.Thesecondfundamentalformcanalsobecastinthefollowingmatrixform:(256)

wherevisadirectionvector,vTisitstranspose,andIISisthesecondfundamentalformtensor.Asindicatedbefore,thenotationregardingthedotproductoperationmaynotbestandardinthecommonlyemployedmatrixnotation.Also,althoughthecoefficientsofthesecondfundamentalformdependonthepositionbutnotthedirection,thesecondfundamentalformdependsonboth.Asseenbefore,thecoefficientsofthesecondfundamentalformsatisfythefollowingrelations:(257)e = b11f = b12 = b21g = b22Hence,thesecondfundamentalformtensorIISisthesameasthesurfacecovariantcurvaturetensorb,thatis:(258)

Asaconsequenceofthepreviousstatements,weseethatthesecondfundamentalformcanalsobegiveninacompacttensornotationformby:(259)IIS = bαβduαduβwheretheindexedbaretheelementsofthesurfacecovariantcurvaturetensorandα, β = 1, 2.ThesecondfundamentalformofthesurfaceatagivenpointandinagivendirectioncanalsobeexpressedintermsofthefirstfundamentalformISandthenormalcurvatureκnofthesurfaceatthatpointandinthatdirection(see§4.2↓)as:(260)IIS = κnIS = κn(ds)2BasedonEqs.259↑and226↑,foraMongepatchoftheformr(u, v) = (u, v, f(u, v)),thesecondfundamentalformisgivenby:(261)IIS = (fuududu + 2fuvdudv + fvvdvdv) ⁄ √(1 + fu2 + fv2)wherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates.Whilethefirstfundamentalformisinvariantunderisometrictransformations,thesecondfundamentalformisinvariantunderrigidmotiontransformations.Thesecondfundamentalformisalsoinvariant(consideringspatially-fixeddirections)underpermissiblesurfacere-parameterizationsthatmaintainthesenseofthenormalvectortothesurface,n.However,thesecondfundamentalformchangesitssignifthesenseofnisreversed.Asindicatedearlier,whilethefirstfundamentalformencompassestheintrinsicgeometryofthesurface,thesecondfundamentalformencompassesitsextrinsicgeometry.Consequently,thefirstfundamentalformisassociatedwiththesurfacecovariantmetrictensor,whilethesecondfundamentalformisassociatedwiththesurfacecovariantcurvaturetensor.Amajordifferencebetweenthetwofundamentalformsisthatwhilethefirstfundamentalformispositivedefinite,asstatedpreviously,thesecondfundamentalformisnotnecessarilypositiveordefinite.Unlikespacecurveswhicharecompletelydefinedbyspecifiedcurvatureandtorsion,κandτ,providingarbitraryfirstandsecondfundamentalformsisnotasufficientconditionfortheexistenceanddeterminationofasurfacewiththeseforms.Thisisduetothefactthatthefirstandsecondfundamentalformswhenindependentlydefineddonotprovideacceptabledeterminationforthesurfaceunlesstheysatisfyextracompatibilityconditionstoensuretheconsistencyoftheseformsandsecuretheexistenceofasurfaceassociatedwiththeseforms.Inmoretechnicalterms,definingsixfunctionsE, F, G, e, f, gofclassC3onasubsetofℝ2wherethesefunctionssatisfytheconditionsforthecoefficientsofthefirstandsecondfundamentalforms(inparticularE, G > 0andEG − F2 > 0)doesnotguaranteetheexistenceofasurfaceoverthegivensubsetwithafirstfundamentalform:E(du1)2 + 2Fdu1du2 + G(du2)2andasecondfundamentalform:e(du1)2 + 2fdu1du2 + g(du2)2.Furthercompatibilityconditionsrelatingthefirstandsecondfundamentalformsarerequiredtofullyidentifythesurfaceandsecureitsexistence.Theabovedifferencebetweencurvesandsurfacesmaybelinkedtothefactthattheconditionsforthecurvesareestablishedbasedontheexistencetheoremforordinarydifferentialequationswheretheseequationsgenerallyhaveasolution,whiletheconditionsforthesurfacesareestablishedbasedontheexistencetheoremforpartialdifferentialequationswhichhavesolutionsonlywhentheymeetadditionalintegrabilityconditions.Thedetailsshouldbesoughtinmoreextensivebooksondifferentialgeometry.Followingthestatementsinthelastparagraphs,therequiredcompatibilityconditionsfortheexistenceofasurfacewithpredefinedfirstandsecondfundamentalformsaregivenbytheCodazzi-Mainardiequations(Eqs.294↓-295↓)plusthefollowingequation:(262)eg − f2 = F(∂uΓ222 − ∂vΓ212 + Γ122Γ211 − Γ112Γ212) + E(∂uΓ122 − ∂vΓ112 + Γ122Γ111 + Γ222Γ112 − Γ112Γ112 − Γ212Γ122)whereallthesymbolsareasdefinedpreviouslyandtheChristoffelsymbolsbelongtothesurface.Fromtheabovestatements,thefundamentaltheoremofspacesurfacesindifferentialgeometry,whichistheequivalentofthefundamentaltheoremofcurves(see§2.3↑),emerges.Thetheoremstatesthat:givensixsufficientlysmoothfunctionsE, F, G, e, f, gonasubsetofℝ2satisfyingthefollowingconditions:1.E, G > 0andEG − F2 > 0,and2.E, F, G, e, f, gsatisfyEqs.262↑and294↓-295↓,thenthereisauniquesurfacewithE, F, Gasitsfirstfundamentalformcoefficientsande, f, gasitssecondfundamentalformcoefficients.Hence,iftwosurfacesmeetalltheseconditions,thentheyareidenticalwithinarigidmotiontransformationinspace.Asaresult,thefundamentaltheoremofsurfaces,likethefundamentaltheoremofcurves,providestheexistenceanduniquenessconditionsforsurfaces.Ontheotherhand,accordingtooneofthetheoremsofBonnet,iftherearetwosurfacesofclassC3,S1:Ω → ℝ3andS2:Ω → ℝ3,whicharedefinedoveraconnectedsetΩ ⊆ ℝ2andtheyhaveidenticalfirstandsecondfundamentalforms,thenthetwosurfacescanbemappedoneachotherbyapurelyrigidmotiontransformation.Theexistenceofthesesurfacesguaranteesthecompatibilityoftheirfirstandsecondfundamentalformsandhencetheyareidenticalwithinarigidmotiontransformationaccordingtothefundamentaltheoremofsurfaces.Itisnoteworthythattwosurfaceshavingidenticalfirstfundamentalformsbutdifferentsecondfundamentalformsmaybedescribedasapplicable.Anexampleofapplicablesurfacesisplaneandcylinder.Althoughallapplicablesurfaces,accordingtothisdefinition,areisometricsincetheyhaveidenticalfirstfundamentalforms,notallisometricsurfacesareapplicablesincetwoisometricsurfacesmayalsohaveidenticalsecondfundamentalformsasitisthecaseoftwoplanes.ItshouldberemarkedthatsomeauthorsuseL, M, Ninsteadofe, f, gtosymbolizethecoefficientsofthesecondfundamentalform.However,theuseofe, f, gisadvantageoussincetheycorrespondnicelytothecoefficientsofthefirstfundamentalformE, F, Gmakingtheformulaeinvolvingthefirstandsecondfundamentalformsmoresymmetricandmemorable.Ontheotherhand,theuseofL, M, Nisalsoadvantageouswhenrecitingformulaeespeciallyiftheformulaecontainthecoefficientsofbothfundamentalforms;moreover,itislesssusceptibletoerrorsinthewritingandtypingoftheseformulae.Anotherremarkisthatthecoefficientgofthesecondfundamentalformshouldnotbeconfusedwiththesymbolgofthedeterminantofthespacecovariantmetrictensorwhichiscommonlyusedintheliteratureoftensorcalculusanddifferentialgeometrybutwedonotuseitinthepresentbook.Thecoefficientfofthesecondfundamentalformshouldalsobedistinguishedeasilyfromthesymbolfwhichiswidelyusedinthemathematicalliteraturetosymbolizemathematicalfunctionsandhencewekeptitsuseinthisbookforthesakeofreadabilitywhilemakingsomeefforttoavoid

potentialconfusion.

3.6.1 DupinIndicatrixOndisplacingthetangentplaneofasurfaceSatagivenpointPinfinitesimallytowardthesurfaceintheorientationofthenormalvectorofSatP,theintersectioncurvesofSwiththedisplacedtangentplanewilltakeaparticularanddistinctiveshapedependingonthelocalshapeofSintheneighborhoodofP(refertoFigs.34↓,35↓and36↓).ThisideaisthebaseforcharacterizingandquantifyingthelocalshapeofasurfaceataparticularpointusingtheconceptofDupinindicatrix(seeFootnote14in§8↓).Dupinindicatrixatagivenpointonasufficientlysmoothsurfaceisafunctionofthecoefficientsofthesecondfundamentalformatthepointandhenceitisafunctionofthesurfacecoordinates.Dupinindicatrixisanindicatorofthedepartureofthesurfacefromthetangentplaneinthecloseproximityofthepointoftangency.Accordingly,thesecondfundamentalformcoefficientsareusedinDupinindicatrixtomeasurethisdeparture.Inquantitativeterms,Dupinindicatrixisthefamilyofconicsectionsgivenbythefollowingquadraticequation:(263)eu2 + 2fuv + gv2 = ±1wheree, f, garethecoefficientsofthesecondfundamentalformatthepointwiththecoordinatesuandv.Asseen,Dupinindicatrix,whichisrepresentedbyanexpressionsimilarinformtothesecondfundamentalform,dependsonthecoefficientsofthesecondfundamentalformandthecoordinatesoftheparticularpointonthesurfaceandhenceitisafunctionofpositionbut,unlikethesecondfundamentalform,itdoesnotdependonthedirection.

Figure34 Contourcurvesonatypicalsurfaceintheneighborhoodofanellipticpoint.

Figure35 Contourcurvesonatypicalsurfaceintheneighborhoodofaparabolicpoint.

Figure36 Contourcurvesonatypicalsurfaceintheneighborhoodofahyperbolicpoint.Asaconsequenceofthepreviousstatements,Dupinindicatrixcanbeusedtoclassifythesurfacepointswithrespecttothelocalshapeofthesurfaceaselliptic,parabolic,hyperbolicorflat(see§4.9↓).AtanellipticpointtheDupinindicatrixisanellipseorcircle(seeFootnote15in§8↓),ataparabolicpointtheDupinindicatrixbecomestwoparallellines,whileatahyperbolicpointtheDupinindicatrixbecomestwoconjugatehyperbolas(seeFig.37↓).AtaflatpointtheDupinindicatrixisnotdefined.MoredetailsaboutDupinindicatrixandthelocalshapeofsurfacewillbegivenin§4.9↓.ItisnoteworthythatinallcaseswheretheDupinindicatrixisdefined,theprincipaldirections(see§4.4↓)atthepointcoincidewiththetwoperpendicularaxesoftheindicatrixasseeninFig.37↓.

Figure37 Dupinindicatrixat(a)ellipticpoint,(b)parabolicpointand(c)hyperbolicpoint.Theprincipaldirections(referto§4.4↓)arelabeledasd1andd2.

3.7 ThirdFundamentalFormThethirdfundamentalformIIISofaspacesurfaceisdefinedby:(264)IIIS = dn⋅dn = cαβduαduβwherenistheunitnormalvectortothesurfaceatagivenpointP,cαβarethecoefficientsofthethirdfundamentalformatPandα, β = 1, 2.Thecoefficientsofthethirdfundamentalformaregiveninfulltensornotationby:(265)cαβ = gijni, αnj, βwheregijisthespacecovariantmetrictensorandtheindexednistheunitnormalvectortothesurface.Wenotethatthesecoefficientsarerealnumbers.Thecoefficientsofthethirdfundamentalformmayalsobegivenby:(266)cαβ = aγδbαγbβδwhereaγδisthesurfacecontravariantmetrictensorandtheindexedbarethecoefficientsofthesurfacecovariantcurvaturetensor.

3.8 RelationshipbetweenFirst,SecondandThirdFundamentalFormsThefirst,secondandthirdfundamentalformsarelinked,throughtheGaussiancurvatureKandthemeancurvatureH(see§4.5↓and4.6↓),bythefollowingrelation:(267)KIS − 2HIIS + IIIS = 0Accordingly,thecoefficientsofthefirst,secondandthirdfundamentalformsarecorrelated,throughtheGaussiancurvatureandthemeancurvature,bythefollowingrelation:(268)Kaαβ − 2Hbαβ + cαβ = 0Infact,Eq.267↑canbeobtainedfromEq.268↑bymultiplyingthelatterbyduαduβandapplyingthesummationconvention.OnmultiplyingbothsidesofEq.268↑byaαβandshiftingtheindicesweobtain:(269)Kaαα − 2Hbαα + cαα = 0thatis:(270)tr(cβα) = 4H2 − 2KThetransitionformEq.269↑toEq.270↑isjustifiedbythefactthat:aαα = δαα = δ11 + δ22 = 2andH = (bαα ⁄ 2)(seeEq.383↓).

3.9 RelationshipbetweenSurfaceBasisVectorsandtheirDerivativesThefocusofthissectionistheequationsofGaussandWeingartenwhich,forsurfaces,aretheanalogueoftheequationsofFrenet-Serretforcurves.WhiletheFrenet-SerretformulaeexpressthederivativesofT, N, Bascombinationsofthesevectorsusingκandτascoefficients,theequationsofGaussandWeingartenexpressthederivativesofE1,  E2,  nascombinationsofthesevectorswithcoefficientsbasedonthefirstandsecondfundamentalforms.Asshownearlier(see§2.2↑),threeunitvectorscanbeconstructedoneachpointatwhichthecurvaturedoesnotvanishofaclassC2spacecurve:thetangentT,theprincipalnormalNandthebinormalB.Thesemutuallyorthogonalvectors(i.e.T⋅N = T⋅B = N⋅B = 0)canserveasasetofbasisvectorsfortheembedding3Dspace.Hence,thederivativesofthesevectorswithrespecttothedistancetraversedalongthecurve,s,canbeexpressedascombinationsofthisset,sincethesederivativesare3Dvectorsthatresideinthisspace,asdemonstratedbytheFrenet-Serretformulae(referto§2.5↑).Similarly,thesurfacevectors:E1 = (∂ r ⁄ ∂u1),E2 = (∂ r ⁄ ∂u2)andtheunitnormalvectortothesurface,n,ateachregularpointonaclassC2surfaceformabasissetfortheembedding3Dspaceandhencetheirpartialderivativeswithrespecttothesurfacecoordinates,u1andu2,canbeexpressedascombinationsofthisset.TheequationsofGaussandWeingartendemonstratethisfact.TheequationsofGaussexpressthepartialderivativesofthesurfacevectors,E1andE2,withrespecttothesurfacecoordinatesascombinationsoftheE1,  E2,  nbasisset,thatis:(271)∂E1 ⁄ ∂u1 = Γ111 E1 + Γ211 E2 + en(272)∂E1 ⁄ ∂u2 = Γ112 E1 + Γ212 E2 + fn = ∂E2 ⁄ ∂u1

(273)∂E2 ⁄ ∂u2 = Γ122 E1 + Γ222 E2 + gnwheree, f, garethecoefficientsofthesecondfundamentalform.Theseequationscanbeexpressedcompactly,withpartialuseoftensornotation,as:(274)∂Eα ⁄ ∂uβ = Γγ

αβEγ + bαβnwhereα, β = 1, 2,theChristoffelsymbolofthesecondkindΓγ

αβisbasedonthesurfacemetric,asgivenbyEqs.73↑-78↑,andbαβisthesurfacecovariantcurvaturetensor.Thelastequationcanbeexpressedinfulltensornotationas:(275)xi

α, β = Γγαβxi

γ + bαβni

wherethesymbolsandnotationsareasdefinedpreviously.Theseequationsmayalsobeexpressedas:(276)xi

α;β = bαβni

wherethecovariantderivativenotationisinuseandarectangularCartesiancoordinatesystemforthespaceisassumed(refertoEq.459↓).Likewise,theequationsofWeingartenexpressthepartialderivativesoftheunitnormalvectortothesurface,n,withrespecttothesurfacecoordinatesascombinationsofthesurfacevectors,E1andE2,thatis:(277)∂n ⁄ ∂u1 = [(fF − eG) ⁄ a]E1 + [(eF − fE) ⁄ a]E2(278)∂n ⁄ ∂u2 = [(gF − fG) ⁄ a]E1 + [(fF − gE) ⁄ a]E2whereE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalformsanda = EG − F2isthedeterminantofthesurfacecovariantmetrictensor.WenotethattheseexpressionsarealsocombinationsoftheE1,  E2,  nbasissetbutwithvanishingnormalcomponents.TheaboveequationsofWeingartencanbeexpressedcompactly,withpartialuseoftensornotation,as:(279)∂n ⁄ ∂uα =  − bα

βEβwherebα

β( = bαγaγβ)isthemixedtypeofthesurfacecurvaturetensor,bαγisthesurfacecovariantcurvaturetensorandaγβisthesurfacecontravariantmetrictensor(refertoEq.223↑andsurroundingtext).Theycanalsobeexpressedwithfulluseoftensornotationas:(280)ni

, α =  − bαγaγβxiβ =  − bα

βxiβ

Tomakesenseofthisequation,thevectorni, αisorthogonaltoniandhenceitisparalleltothetangentspaceofthe

surface,soitcanbeexpressedasalinearcombinationofthesurfacebasisvectorsxiβ,thatis:

ni, α = dα

βxiβ

foracertainsetofcoefficientsdα

β =  − bαγaγβ,asseenabove.TheWeingartenequationsmayalsobeexpressedinmatrixformas:(281)

whereIISisthesurfacecovariantcurvaturetensorandIS − 1isthesurfacecontravariantmetrictensor.Infact,thisis

thematrixformofEq.279↑.Thepartialderivativesoftheunitnormalvectortothesurface,n,withrespecttothesurfacecoordinates,u1andu2,arelinkedtotheGaussiancurvatureK(see§4.5↓)andthesurfacebasisvectors,E1andE2,aswellasthecoefficientsofthefirstandsecondfundamentalformsE, F, G, e, f, gbythefollowingrelation:(282)(∂n ⁄ ∂u1) × (∂ n ⁄ ∂u2) = [(eg − f2) ⁄ (EG − F2)]( E1 ×  E2) = K(E1 ×  E2)whereEq.356↓isusedinthelaststep.Thepartialderivativesoftheunitnormalvectortothesurface,n,withrespecttothesurfacecoordinates,u1andu2,arealsolinkedtotheGaussianandmeancurvatures,KandH,andthecoefficientsofthefirstandsecondfundamentalformsE, F, G, e, f, gbythefollowingrelations:(283)(∂n ⁄ ∂u1)⋅(∂ n ⁄ ∂u1) = 2eH − EK(284)(∂n ⁄ ∂u1)⋅(∂ n ⁄ ∂u2) = 2fH − FK(285)(∂n ⁄ ∂u2)⋅(∂ n ⁄ ∂u2) = 2gH − GKTheaboveequationsofWeingarten(Eqs.277↑-278↑)canbesolvedforthesurfacebasisvectors,E1andE2,andhencethesevectorscanbeexpressedascombinationsofthepartialderivativesofthenormalvector,n,thatis:(286)E1 = [(fF − gE) ⁄ b](∂n ⁄ ∂u1) + [(fE − eF) ⁄ b](∂n ⁄ ∂u2)(287)E2 = [(fG − gF) ⁄ b](∂n ⁄ ∂u1) + [(fF − eG) ⁄ b](∂n ⁄ ∂u2)whereb = eg − f2isthedeterminantofthesurfacecovariantcurvaturetensor.FromEq.274↑,itcanbeseenthatthecoefficientsofthesurfacecovariantcurvaturetensor,bαβ,aretheprojectionsofthepartialderivativeofthesurfacebasisvectors,∂Eα ⁄ ∂uβ,inthedirectionoftheunitnormalvectortothesurface,n,thatis:(288)bαβ = (∂Eα ⁄ ∂uβ)⋅nThiscanalsobeseendirectlyfromthedefinitionofthecoefficientsofthesecondfundamentalform,i.e.Eqs.249↑-251↑.Asindicatedabove,theessenceoftheaboveequationsofGaussandWeingartenisthatthepartialderivativesofE1,E2andncanberepresentedascombinationsofthesevectorswithcoefficientsobtainedfromthecoefficientsofthefirstandsecondfundamentalformsandtheirpartialderivatives.Theaboveequationsarevariousdemonstrationsofthisfact.ForaMongepatchoftheformr(u, v) = (u, v, f(u, v)),theGaussequationsaregivenby:(289)∂E1 ⁄ ∂u = [fufuuE1 + fvfuuE2 + fuu√(1 + fu2 + fv2) n] ⁄ (1 + fu2 + fv2)(290)∂E1 ⁄ ∂v = [fufuvE1 + fvfuvE2 + fuv√(1 + fu2 + fv2) n] ⁄ (1 + fu2 + fv2) = ∂ E2 ⁄ ∂u(291)∂E2 ⁄ ∂v = [fufvvE1 + fvfvvE2 + fvv√(1 + fu2 + fv2) n] ⁄ (1 + fu2 + fv2)wherethesubscriptsuandvrepresentpartialderivativesoffwithrespecttothesurfacecoordinatesuandv.Similarly,foraMongepatchoftheformr(u, v) = (u, v, f(u, v)),theWeingartenequationsaregivenby:(292)∂n ⁄ ∂u = [(fufvfuv − fuufv2 − fuu) ⁄ (1 + fu2 + fv2)3 ⁄ 2] E1 + [(fufvfuu − fu2fuv − fuv) ⁄ (1 + fu2 + fv2)3 ⁄ 2] E2(293)∂n ⁄ ∂v = [(fufvfvv − fuvfv2 − fuv) ⁄ (1 + fu2 + fv2)3 ⁄ 2] E1 + [(fufvfuv − fu2fvv − fvv) ⁄ (1 + fu2 + fv2)3 ⁄ 2] E2

3.9.1 Codazzi-MainardiEquationsFromtheaforementionedequationsofGaussandWeingarten,supportedbyfurthercompatibilityconditions,thefollowingequations,calledCodazziorCodazzi-Mainardiequations,canbederived:(294)(∂b12 ⁄ ∂u1) − (∂b11 ⁄ ∂u2) = b22Γ211 − b12(Γ212 − Γ111) − b11Γ112(295)(∂b22 ⁄ ∂u1) − (∂b21 ⁄ ∂u2) = b22Γ212 − b12(Γ222 − Γ112) − b11Γ122wheretheChristoffelsymbolsarebasedonthesurfacemetric.Theseequationscanbeexpressedcompactlyintensornotationas:(296)(∂bαβ ⁄ ∂uγ) − (∂bαγ ⁄ ∂uβ) = bδβΓδ

αγ − bδγΓδαβ

Now,ifwearrangethetermsofthelastequationandsubtractthetermbαδΓδγβfrombothsidesweobtain:

(297)(∂bαβ ⁄ ∂uγ) − bδβΓδαγ − bαδΓδ

γβ = (∂bαγ ⁄ ∂uβ) − bδγΓδαβ − bαδΓδ

γβwhichcanbeexpressedcompactly,usingthecovariantderivativenotation(see§7↓),as:(298)bαβ;γ = bαγ;βTheCodazzi-MainardiequationsintheformgivenbyEq.298↑revealthatthereareonlytwoindependentcomponentsfortheseequationsbecause,addingtothefactthatalltheindicesrangeover1and2andhencewehaveonlyeightcomponents,thecovariantderivativeaccordingtoEq.298↑issymmetricinitslasttwoindices(i.e.βandγ),andthecovariantcurvaturetensorissymmetricinitstwoindices(i.e.bαβ = bβα).Thesetwoindependentcomponentsaregivenby:(299)bαα;β = bαβ;αwhereα ≠ βandthereisnosummationoverα.Onwritingtheseequationsinfull,usingthecovariantderivativeexpression[i.e.bαβ;γ = (∂bαβ ⁄ ∂uγ) − bδβΓδ

αγ − bαδΓδβγ]andnotingthatonetermofthecovariantderivativeexpressionis

thesameonbothsidesandhenceitdropsaway,wehave:(300)(∂bαα ⁄ ∂uβ) − bαδΓδ

αβ = (∂bαβ ⁄ ∂uα) − bδβΓδαα

whereα ≠ βwithnosumonα.Itshouldberemarkedthatasaconsequenceoftheaforementionedtwosymmetries,thecovariantderivativeofthesurfacecovariantcurvaturetensor,bαβ;γ,isfullysymmetricinallofitsindices.Thereisalsoanothermoregeneralequationcalled(accordingtosomeauthors)theGauss-Codazziequationwhichisgivenby:(301)Rδ

αβγxiδ = xi

δbδβbαγ − xi

δbδγbαβ + nibαβ;γ − nibαγ;β

ThetangentialcomponentofthisequationrepresentsTheoremaEgregium(intheformgivenbyEq.228↑)whileitsnormalcomponentrepresentstheCodazziequation(intheformgivenbyEq.298↑).Thereaderisadvisedtoreferto§4.7↓abouttheessenceofTheoremaEgregiumasanexpressionofthefactthatcertaintypesofcurvatureareintrinsicpropertiestothesurfaceandhencetheycanbeexpressedintermsofpurelyintrinsicparametersobtainedfromthefirstfundamentalform.

3.10 SphereMappingSpheremappingorGaussmappingisacorrelationbetweenthepointsofasurfaceandtheunitspherewhereeachpointonthesurfaceisprojectedontoitsunitnormalasapointontheunitspherewhichiscenteredattheoriginofcoordinates.Thissortofmappingforsurfacesissimilartothesphericalindicatrixmapping(see§5.5↓)forspacecurves.Intechnicalterms,letSbeasurfaceembeddedinanℝ3spaceandS1representstheorigin-centeredunitsphereinthisspace,thenGaussmappingisgivenby:(302){N:S → S1, N(P) = P}wherethepointP(x, y, z)onthetraceofSismappedbyNontothepointP(x, y, ž)onthetraceoftheunitspherewithx, y, zbeingthecoordinatesofPandx, y, žbeingthecoordinatesoftheorigin-basedpositionvectorofthenormalvectortothesurface,n,atP.Tohaveasingle-valuedspheremapping,thefunctionalrelationrepresentingthesurfaceSshouldbeone-to-one.TheimageDontheunitsphereofaGaussmappingofapatchDonasurfaceSiscalledthesphericalimageofD.ThelimitoftheratiooftheareaofaregionZonthesphericalimagetotheareaofthecorrespondingregionZonthesurfaceSintheneighborhoodofagivenpointPonSequalstheabsolutevalueoftheGaussiancurvature|K|atPasZshrinkstothepointP,thatis:(303)

whereσstandsforarea,andKPistheGaussiancurvatureatP.ThetendencyofZtoPshouldbeunderstoodinthegivensense.AtagivenpointPonasurface,wheretheGaussiancurvatureisnon-zero,thereexistsaneighborhoodNofPwhereaninjectivemappingcanbeestablishedbetweenNanditssphericalimageÑ.Aconformalcorrespondencecanbeestablishedbetweenasurfaceanditssphericalimageiffthesurfaceisasphereoraminimalsurface(see§6.7↓).Weshouldremarkthatalthoughtheterm“sphericalimage”isrelatedtosurfacesinthecontextofspheremapping,itcanalsobeusedforcurvessincecurvescanalsohavesphericalimagesinawelldefinedsense.

3.11 GlobalSurfaceTheoremsInthissmallsectionwestate,withinthefollowingbulletpoints,afewglobaltheoremsrelatedtosurfacestohaveatasteofthisfieldofdifferentialgeometryofsurfaceswhoseinvestigationisnotthemainobjectiveofthepresentbook.•PlanesaretheonlyconnectedsurfacesofclassC2whoseallpointsareflat.•SpheresaretheonlyconnectedclosedsurfacesofclassC3whoseallpointsaresphericalumbilical(see§4.10↓).•SpheresaretheonlyconnectedcompactsurfacesofclassC3withconstantGaussiancurvature.•SpheresaretheonlyconnectedcompactsurfaceswithconstantmeancurvatureandpositiveGaussiancurvature.•Thetangentplanetoacylinderoraconeisconstantalongtheirgenerators.•Anycompactsurfaceinℝ3shouldhavepointswithpositiveGaussiancurvature.•Anycompactsurfaceinℝ3,excludingsphere,shouldhavepointswithnegativeGaussiancurvature.•Thereaderisalsoreferredto§4.8↓fortheglobalformoftheGauss-Bonnettheorem.

3.12 ExercisesExercise1.Givethemathematicaldefinitionofspacesurfaceandexplainthedifferencebetweenasurfaceanditstraceaccordingtothisdefinition.Exercise2.Whatisthemathematicalconditionforasurfacetoberegularataparticularpointintermsofitsbasisvectors?Exercise3.Statethethreemainmathematicalmethodsfordefiningaspacesurfaceandcomparethemexplaininganyadvantagesordisadvantagesinusingoneofthesemethodsortheothersinvariouscontexts.Exercise4.Classifythethreemethodsofthelastquestionintotwomaincategoriesanddiscussthesecategories(see§1.4.3↑).Exercise5.What“coordinatepatchofclassCn”means?Whatarethemathematicalconditionsthatshouldbesatisfiedbysuchapatch?Exercise6.ShowthataMongepatchoftheformr(u, v) = (u, v, f(u, v))isregularofclassCniffisofthisclass.Exercise7.Givearigorousdefinitionof“tangentvector”ofasurfacecurveataparticularpointonthesurface.Exercise8.Howthetangentplaneofasurfaceataparticularpointisrelatedtothebasisvectorsofthesurfaceatthatpoint?Exercise9.Findtheequationofthetangentplanetotheellipsoidwhichisrepresentedparametricallyby:r(θ, φ) = (2sinθcosφ, 1.5sinθsinφ, 0.5cosθ)atthepointwithθ = 1.3andφ = 0.72.Exercise10.Findtheequationofthetangentplaneofasurfacerepresentedparametricallyby:r(u, v) = (6v, 2u, 1.4u2 + 6)atthepointwithu = 1.2andv = 3.6.Exercise11.Showthatforacircularconethetangentplaneatallpointsofanyoneofitsgeneratorsisthesame.Exercise12.Discussthefollowingstatementexplainingitsmeaninginsimplewords:“ThetangentspaceataspecificpointPofasurfaceisapropertyofthesurfaceatPandhenceitisindependentofthepatchthatcontainsP”.Exercise13.Doesthetangentspaceofasurfaceatagivenpointdependontheparticularparameterizationofthesurface?Exercise14.Findtheequationoftheplanepassingthroughthepoint( − 1, 3,  − 9)andspannedbythetwovectors(3, 0.5, 1.2)and(0.9, 3, 6.8).Exercise15.Define,mathematically,thenormalunitvectornofasurfaceatagivenpointintermsofthetwobasisvectorsofthesurfaceatthatpoint.Exercise16.Calculate,symbolically,thenormalunitvectornatageneralpointofasurfacedefinedby:S(x, y) = (x, y, f)wheref = f(x, y)isadifferentiablefunction.Exercise17.Findtheequationofthenormallineofahyperboloidofonesheetgivenby:r(ξ, θ) = (1.6coshξcosθ, 2.1coshξsinθ, 0.4sinhξ)atthepointwithξ = 3.2andθ = 1.5.Exercise18.Findtheequationofthenormallineofasurfacerepresentedby:r(u, v) = (3u, u2 + v, 5v)atthepointwithu = 2.5andv =  − 1.8.Exercise19.Define“Mongepatch”givingitsthreeforms.Whichoftheseformsisthemostcommoninuse?Exercise20.Define,briefly,thefollowingtermswithsomeexamplesrepresentingtheseconcepts:simplesurface,simplyconnectedregiononasurface,closedsurface,compactsurface,elementarysurface,orientedsurfaceanddevelopablesurface.Exercise21.Whichofthefollowingisasimplesurfaceandwhichisnot:cylinder,hyperboloidoftwosheets,torus,Kleinbottle,andellipticparaboloid?Exercise22.Isthesurfacerepresentedbytheequation:x2 + y2 + z2 = 9compact?Whataboutthesurfacerepresentedby:x2 − y2 − z2 = 4?Exercise23.WhytheMobiusstripisnotanorientablesurface?Giveanotherexampleofanon-orientablesurface.Exercise24.Whatisconformalmapping?Whatisdirectandinverseconformalmapping?Exercise25.Describestereographicmappingmakingasimplesketchrepresentingthistype

ofmapping.Exercise26.Provethatstereographicmappingisconformal.Exercise27.Defineisometricmappinggivinganexampleofsuchmappingbetweentwotypesofsurface.Exercise28.Whatarethemathematicalconditionsfortwosurfacestobeisometric?Doesisometryrelatetotheintrinsicorextrinsicpropertiesofthesurfaceandwhy?Exercise29.Whatisthedistinctivepropertyofanisometricrelationbetweentwosurfacesintermsofangles,arclengthsandareasdefinedonthetwosurfaces?Exercise30.Whatistherelationbetweenconformalmappingandisometricmapping?Exercise31.Whatislocalisometry?Whatisthedifferencebetweenlocalisometryandglobalisometry?Exercise32.ShowthatEq.160↑appliestolocalisometricmapping.Exercise33.AsurfaceS1ismappedisometricallyontoanothersurfaceS2.HowtheintrinsicpropertiesofS1willbeaffectedbythismapping?Exercise34.Makeacleardistinctionbetweenthetangentsurfaceofacurveandthetangentplaneofasurfacedescribingeachofthesebriefly.Exercise35.Whatisthemeaningof“branchofthetangentsurfaceofacurveCatagivenpointPonthecurve”?Exercise36.Giveabriefdefinitionofinvoluteandevolute.Exercise37.WritedownageneralmathematicalrelationrepresentingthepositionvectorofapointPonaspacesurfaceasafunctionofthesurfacecoordinates,u1andu2,wherethesurfaceisembeddedina3DEuclideanspacecoordinatedbyarectangularCartesiansystem.Exercise38.Describe,indetail,howacoordinategridisconstructedonaspacesurfacewithacleardefinitionofthecoordinatecurvesusedtobuildthisgrid.Exercise39.Makeasimpleandfullylabeledsketchofaspacesurfacecoordinatedbyacurvilineargridwiththetwocovariantsurfacebasisvectorsandtheunitnormalvectortothesurfaceatonepoint.Exercise40.Asurfaceisrepresentedspatiallyby:r(u, v) = (u, 5, 2v).Discussthetypeandthemainpropertiesofthissurface.Exercise41.Define,symbolically,thecovariantbasisvectorsofaspacesurfaceintermsofthecoordinatesoftheambientspacexiandthecoordinatesofthesurfaceuα.Exercise42.GivethesymbolusedintensornotationtorepresentthesurfacebasisvectorsEα.Fromanalyzingthissymbol,describehowthesurfacebasisvectorscanberegardedascovariantandcontravariantvectorsatthesametime.Exercise43.Write,infulltensornotation,theequationrepresentingthecovariantformoftheunitnormalvectortoaspacesurface.Exercise44.AlthoughE1andE2arelinearlyindependentattheregularpointsofthesurfacetheyarenotnecessarilyorthogonalorofunitlength.IsitpossibletoconstructanorthonormalsetofbasisvectorsfromE1andE2?Ifso,how?Exercise45.Followingatransformationfromanunbarredsurfacecoordinatesystemtoabarredsurfacecoordinatesystem,whatarethemathematicalexpressionsrepresentingthebarredbasisset,Ẽ1andẼ2,intermsoftheunbarredbasisset,E1andE2?Givetheseexpressionsinvectorandtensornotations.Exercise46.Define,descriptivelyandmathematically,thesetofcontravariantbasisvectorsforaspacesurfacediscussinghowthesevectorscanberegardedascovariantandcontravariantvectorssimultaneously.Exercise47.Howthecovariantandcontravariantbasissetsofaspacesurfacecanbeobtainedfromeachother?Statethisinwordsandmathematicallydefiningallthesymbolsinvolved.Exercise48.WhatisthesignificanceofthefollowingrelationinvolvingthecovariantandcontravariantsurfacebasisvectorsandtheKroneckerdelta:Eα⋅Eβ = δαβ?Exercise49.Define,mathematically,thecoefficientsofthesurfacemetrictensorintermsofthesurfacecovariantbasisvectorsinEuclideanandRiemannianspaces.Exercise50.Givethefundamentalrelationthatprovidestheimportantlinkbetweenthemetrictensorofasurfaceandthemetrictensorofitsenvelopingspace.Exercise51.Findthesurfacebasisvectors,E1andE2,andthecoefficientsofthefirstfundamentalformofasurfaceparameterizedby:r(u, v) = (aucosv, businv, cu2)wherea, b, careconstants.Exercise52.Find,symbolically,thefirstfundamentalformofacylinderrepresentedby:r(u, v) = (f1(u), f2(u), v)

wheref1andf2arecontinuousfunctionsofthegivencoordinate.Exercise53.Giventhefactthatfor3DCartesiansystems:IS = (dx1)2 + (dx2)2 + (dx3)2plusthetransformationequationsfromsphericaltoCartesiancoordinatesin3D,deriveISforsphericalcoordinatesystems.Exercise54.Giventhefactthatfor3DCartesiansystems:IS = (dx1)2 + (dx2)2 + (dx3)2plusthetransformationequationsbetweengeneralcurvilinearandCartesiancoordinatesystemsin3D,provethatforgeneralcurvilinearsystems:IS = aαβduαduβ.Exercise55.Define,mathematically,eachofthefollowingvectors:x1α, x2α, x3α, xi1, xi2.Also,discusstheirattributesasspaceandsurfacevectorsandtheirvariancetype.Exercise56.Discusshowasurfacevectorcanalsobeconsideredasaspacevectorstatingthemathematicallinkbetweenthesurfaceandspacerepresentations.Aretheserepresentationsequivalent?Ifso,how?Exercise57.Writedown,usingtensornotation,themathematicalrelationthatcorrelatesthesurfacebasisvectorstotheunitnormalvectortothesurface.Exercise58.WhatisthesignificanceofthefollowingrelationwhichinvolvesthesurfacecontravariantandcovariantmetrictensorsandtheKroneckerdelta:aαγaγβ = δαβ?Exercise59.Givethematrix[aαβ]thatrepresentsthecontravariantformofthesurfacemetrictensorintermsofthecoefficientsofthefirstfundamentalform.Exercise60.Givethematrix[aαβ]thatrepresentsthemixedformofthesurfacemetrictensor.Exercise61.Writedowntherelationsthatrepresentthetransformationbetweenunbarredandbarredsurfacecoordinatesystems.Exercise62.Howthedeterminantsofthesurfacemetrictensoroftwotransformedcoordinatesystemsofagivensurfacearelinked?Exercise63.ExpresstheChristoffelsymbolsofthefirstkind[αβ, γ]forasurfaceintermsofthesurfacecovariantbasisvectorsandtheirpartialderivatives.Exercise64.Derivethemathematicalrelationbetweenthepartialderivativeofthesurfacemetrictensor∂αaβγandtheChristoffelsymbolsofthefirstkindforthesurface.Exercise65.Howthecoefficientsofthesurfacemetrictensorwillbeaffectedbyscalingthesurfaceupordownbyaconstantpositivescalarfactor?Exercise66.GivethecovariantandcontravarianttypesofthesurfacemetrictensorforaMongepatchoftheformr(u, v) = (u, v, f(u, v)).Exercise67.Discusshowtheconceptof“lengthofstraightsegment”isextendedtothelengthofapolygonalarc.Alsodiscusshowtheconceptof“lengthofpolygonalarc”isextendedtothelengthofanarcofatwistedspacecurve.Exercise68.Derivethefollowingrelationwhichlinksthelengthofanelementofarcofacurveresidingona2Dsurfacetothecovariantmetrictensorofthesurface:(ds)2 = aαβduαduβ.Exercise69.Isthelengthofasurfacecurveanintrinsicorextrinsicpropertyandwhy?Exercise70.Givetheformulaforthelength,L,ofasegmentofat-parameterizedsurfacecurveintermsofthecoefficientsofthefirstfundamentalformofthesurface.Exercise71.Usingthemetrictensor,verifythefollowingrelationwherefrepresentsaMongepatchoftheformr(u, v) = (u, v, f(u, v)):ds = √([1 + fu2]dudu + 2fufvdudv + [1 + fv2]dvdv)Exercise72.Developananalyticalexpressionforthelengthofanelementofarc,ds,ofthecatenaryparameterizedbyEqs.28↑-29↑.Exercise73.Discusshowtheconceptof“areaofpolygonalplanefragment”isextendedtotheareaofasurfaceconsistingofpolygonalplanefragments.Alsodiscusshowtheconceptof“areaofsurfacemadeofpolygonalplanefragments”isextendedtotheareaofageneralizedtwistedspacesurface.Exercise74.Derivethemathematicalexpressionfortheareaofaninfinitesimalelementofasurfaceandtheexpressionfortheareaofasurfacepatch.Exercise75.DeriveEq.208↑forthesurfaceareaofaMongepatchoftheformr(u, v) = (u, v, f(u, v)).Exercise76.Aconeisrepresentedparametricallyina3Dspaceby:r(ρ, φ) = (ρcosφ, ρsinφ, cρ)whereρ, φarepolarcoordinates(ρ ≥ 0and0 ≤ φ < 2π)andcisapositiveconstant.Findthe

areaofthepartoftheconecorrespondingto0 ≤ ρ ≤ AwhereAisagivenpositiveconstant.Exercise77.Derivethemathematicalexpression:cosθ = gijAiBjfortheangleθbetweentwounitsurfacevectors,AandB,wheregijisthespacecovariantmetrictensor.AlsogivethemathematicalexpressionofsinθfortheanglebetweenAandB.Exercise78.Givetwomathematicalexpressionsforthecoefficientsofthesurfacecovariantcurvaturetensorbαβ.Exercise79.Usingasphericalcoordinatesrepresentation,findthedeterminantofthesurfacecurvaturetensorofasphere.Exercise80.Showthat∂βEα = ∂αEβ.Exercise81.Provethatiftwosurfaces,S1andS2,aremappedisometricallyoneontheotherthentheyhaveidenticalfirstfundamentalformcoefficientsattheircorrespondingpoints,i.e.E1 = E2,F1 = F2andG1 = G2.Exercise82.Givethematrix[bαβ]whichrepresentsthecontravariantformofthesurfacecurvaturetensorintermsofthecoefficientsofthefirstandsecondfundamentalforms.Exercise83.Discussandjustifytherelationb = J2bexplainingallthesymbolsinvolved.Exercise84.Whataretheothersymbolsusedbysomeauthorstolabelthecoefficientsofthesecondfundamentalforme, f, g?Discusstheadvantagesanddisadvantagesofusingeachoneofthesesetsofsymbols.Also,writethemathematicalexpressionforthesecondfundamentalformusingthealternativesymbols.Exercise85.Howcanweobtainthemixedformofthesurfacecurvaturetensorbαβfromthecovariantformofthistensorbαβ?Exercise86.ExpressthemeancurvatureHandtheGaussiancurvatureKofasurfaceintermsofthemixedformofthesurfacecurvaturetensorbαβ.Exercise87.Explain,indetails,allthesymbolsandnotationsinvolvedinthefollowingrelation:bαγbβδ − bαδbβγ = (∂Γαβδ ⁄ ∂γ) − (∂Γαβγ ⁄ ∂δ) + ΓωβδΓαωγ − ΓωβγΓαωδExercise88.WritedownthematrixformofthesurfacecovariantcurvaturetensorforaMongepatchoftheformr(u, v) = (u, v, f(u, v)).Exercise89.WhatistherelationbetweentheRiemann-Christoffelcurvaturetensorofthesecondkindandthecurvaturetensorofasurface?Exercise90.GiveallthecoefficientsoftheRiemann-Christoffelcurvaturetensorandthecurvaturetensorforaplanesurface.Exercise91.Onaspacesurface,howmanyindependentnon-vanishingcomponentstheRiemann-Christoffelcurvaturetensorpossesses?Exercise92.Explaintherelationbetweenthecoefficientsofthemetrictensorofasurfaceanditsfirstfundamentalformstatingthenecessaryequations.Exercise93.DerivethemathematicalformulaforISintermsofthecoefficientsofthefirstfundamentalform.Exercise94.Expressthedeterminantofthesurfacecovariantmetrictensorasafunctionofthecoefficientsofthefirstfundamentalform.Exercise95.ExpressE, F, Gasdotproductsofthecovariantbasisvectorsofthesurfaceandrelatethistothespacemetrictensor.Exercise96.Doesthefirstfundamentalformprovideauniquecharacterizationofthespacesurfaceasseeninternallybya2Dinhabitant?Explainwhy.Exercise97.Doesthefirstfundamentalformprovideauniquecharacterizationofthespacesurfaceasseenfromtheexternalambientspace?Explainwhy.Exercise98.Statethemathematicalconditionsfortheprovisionofpositivedefinitenessofthefirstfundamentalform.Exercise99.Givethemathematicalconditionsthatapplytothecoefficientsofthecovariantmetrictensoroftwoisometricsurfacesattheircorrespondingpoints.Exercise100.Explainhowthesecondfundamentalformcharacterizesthesurfacefromtheambientspaceperspectiveandhowtheunitnormalvectortothesurfaceisemployedinthischaracterization.Exercise101.Expressthedeterminantofthesurfacecovariantcurvaturetensorasafunctionofthecoefficientsofthesecondfundamentalform.Exercise102.DerivethemathematicalrelationofthesecondfundamentalformIISintermsofthecoefficientse, f, g.Alsoprovidethemainmathematicaldefinitionsforthecoefficientse, f, gintermsofthesurfacebasisvectorsandtheunitnormalvectortothesurface.Exercise103.WhatistherelationbetweenthesecondfundamentalformIISandthesecondorderdifferentialofthepositionvectord2 rofasurface?Exercise104.Statethemathematicalrelationsbetweenthecoefficientsofthesecond

fundamentalformandthecoefficientsofthesurfacecovariantcurvaturetensor.Exercise105.Express,infulltensornotation,thesecondfundamentalformintermsofthecoefficientsofthesurfacecovariantcurvaturetensorandlinkthistotheexpressioninvolvingthecoefficientse, f, g.Exercise106.Explainhowthefollowingrelationprovidesabridgebetweenthefirstandsecondfundamentalforms:IIS = κnIS.Exercise107.DerivethefollowingrelationwherefisafunctionalrepresentationofaMongepatchoftheformr(u, v) = (u, v, f(u, v)):IIS = (fuududu + 2fuvdudv + fvvdvdv) ⁄ √(1 + fu2 + fv2)Exercise108.Discusshowthefirstandsecondfundamentalformsrepresenttheintrinsicandextrinsicgeometryofthesurface.Exercise109.Iftwosurfaceshaveidenticalfirstandsecondfundamentalforms,shouldtheybecongruent?Exercise110.Whatarethecompatibilityconditionslinkingthefirstandsecondfundamentalformswhichareneededtofullyidentifyasurfaceassociatedwithspecificfirstandsecondfundamentalformsandsecureitsexistence?Exercise111.State,usingmathematicaltechnicalterms,thefundamentaltheoremofspacesurfaces.Exercise112.GiveabriefdefinitionofDupinindicatrixandstateitssignificanceandusageindifferentialgeometry.Exercise113.WhatistheshapeofDupinindicatrixatelliptic,parabolicandhyperbolicpointsonasmoothsurface?WhatistheshapeofDupinindicatrixatflatpoints?Exercise114.MakeasimplesketchtoillustrateDupinindicatrixatanellipticpoint,aparabolicpointandahyperbolicpointonasurfacemarkingthetwoprincipaldirectionsineachcase.Exercise115.WritedownthemathematicalexpressionforthethirdfundamentalformIIISintermsoftheunitnormalvectortothesurfaceandintermsofthecoefficientscαβ.Exercise116.Expressthecoefficientsofthethirdfundamentalformasafunctionofthecoefficientsofthesurfacemetricandcurvaturetensors.Exercise117.Explainallthesymbolsinvolvedinthefollowingequation:KIS − 2HIIS + IIIS = 0.Exercise118.DerivetheequationinthelastquestionusingtheWeingartenequations.Exercise119.Startingfromtheequation:Kaαβ − 2Hbαβ + cαβ = 0,derive,withfullexplanation,thefollowingrelation:tr(cβα) = 4H2 − 2K.Exercise120.ExplainthecorrespondencebetweentheFrenet-SerretformulaeforspacecurvesandtheequationsofGaussandWeingartenforspacesurfaces.Exercise121.Whythepartialderivativesofthesurfacebasisvectors,E1andE2,andtheunitnormalvectortothesurface,n,withrespecttothesurfacecoordinates,u1andu2,canbeexpressedascombinationsofthesevectors?Exercise122.ProveEq.282↑usingtheWeingartenequations.Exercise123.Writethederivativesofthesurfacebasisvectors(i.e.∂βEα)intermsofthesurfacevectors(i.e.Eγandn)intheirvectorandtensorforms.Exercise124.WhatistheessenceoftheequationsofWeingarten?Providequalitativeandquantitativedescriptionsoftheseequations.Alsowritetheseequationsinamatrixforminvolvingthecovariantcurvaturetensorandthecontravariantmetrictensorofthesurface.Exercise125.DeriveWeingartenequationsforaMongepatchoftheformr(u, v) = (u, v, f(u, v)).Exercise126.ExpressthepartialderivativesofnwithrespecttothesurfacecoordinatesintermsoftheGaussianandmeancurvaturesandthecoefficientsofthefirstandsecondfundamentalforms.Exercise127.GivetheequationsofGaussforaMongepatchoftheformr(u, v) = (u, v, f(u, v)).Exercise128.State,usingfulltensornotation,theequationsofCodazzi-Mainardiexplainingallthesymbolsinvolved.Exercise129.Derive,usingtheCodazzi-Mainardiequations,thefollowingrelation:bαβ;γ = bαγ;β.Exercise130.ExplainhowtherelationinthepreviousquestionindicatesthatthereareonlytwoindependentcomponentsfortheCodazzi-Mainardiequations.Whatarethesetwoindependentcomponents?

Exercise131.Describespheremappinginqualitativeandtechnicalterms.Also,explainthemeaningofthefollowingequation:

Exercise132.Provethetheoremrepresentedbytheequationinthepreviousquestion.Exercise133.ProvethatatagivenpointPonasurfacewithK ≠ 0,thereexistsaneighborhoodNofPwhereaninjectivemappingcanbeestablishedbetweenNanditssphericalimageÑ.Exercise134.Stateoneoftheglobaltheoremsofspacesurfaceandexplainwhyitisglobal.

Chapter4Curvature“Curvature”isapropertyofbothcurvesandsurfacesatagivenpointwhichisdeterminedbytheshapeofthecurveorsurfaceatthatpoint.TherearealsoglobalcharacteristicsofcurvatureliketotalcurvatureKt(see§4.8↓)ofasurfacebuttheyarebasedingeneralonthelocalcharacterizationofcurvatureatindividualpoints.Thecurvaturealsohasintrinsicaswellasextrinsicattributesandhenceitcharacterizesthemanifoldinternallyasseenbyaninhabitantofthemanifoldandexternallyasseenbyanoutsider.Inthischapter,weinvestigatethispropertyinitsgeneralmeaningandexaminethemainparametersusedtodescribecurvatureandquantifyitfocusingonspacesurfacesandcurvesembeddedinsuchsurfaces.Thematerialsarelargelybasedona3DflatambientspacecoordinatedbyaCartesianorthonormalsystem.

4.1 CurvatureVectorAtagivenpointPonasurfaceS,aplanecontainingthevectorn,whichisthenormalunitvectortothesurfaceatP,intersectsthesurfaceinasurfacecurveChavingatangentvectortatP.ThecurveCiscalledthenormalsectionofSatPinthedirectionoft.TheprincipalnormalvectorNofanormalsectionatPiscollinearwiththeunitnormalvectornatP.Wenotethatforanormalsection,Nandncanbeparalleloranti-parallelsinceonanorientablesurfacethevectorncanhaveoneoftwopossibledirections.Ontheotherhand,asurfacecurvepassingthroughPinthedirectionoftmaynotbeanormalsectionandhencethevectorsNandnatPhavedifferentorientations.InthiscontextweremarkthatweareconsideringherethepartofthecurveintheneighborhoodofthepointPaspartofanormalsectionornot,andnotnecessarilythewholecurve.Asexplainedbefore,aspacecurveCcanbeparameterizedbys,representingthedistancetraversedalongC,andhencethecurveisdefinedbythepositionvectorr(s).AtagivenpointPonC,thevectorT = (dr ⁄ ds)isaunitvectortangenttoCatPinthedirectionofincreasings.WhenCisembeddedinasurface,TwillbecontainedinthetangentplanetothesurfaceatP,asexplainedpreviouslyin§2.2↑and3.1↑.WenotethatthevectorTcanbeparalleloranti-paralleltotheaforementionedvectort.WechosetointroducetanddefineitinthiswaytobemoregeneralsincethecurveorientationcanbeinonedirectionortheotherandhenceTandtcanbeparalleloranti-parallel.Moreover,tisnotnecessarilyofunitlengthorbasedonanaturalparameterizationofthecurve.ThecurvaturevectorofCatP,whichisorthogonaltoT,isdefinedby:(304)K = dT ⁄ dswhereK,whichistheuppercaseGreekletterkappa,symbolizesthecurvaturevector.ThecurvatureκofCatP(whichisdefinedpreviouslyin§2.2↑)isthemagnitudeofthecurvaturevector,thatis:κ = |K|,andtheradiusofcurvaturewhenκ ≠ 0isitsreciprocal,i.e.Rκ = 1 ⁄ κ,whichistheradiusoftheosculatingcircleofCatP(see§2.6↑).Thecurvaturevectorcanthereforebeexpressedas:(305)K = |K|(K ⁄ |K|) = κNwhereNistheprincipalnormalvectorofthecurveCatPasdefinedpreviouslyin§2.2↑.Thecurvaturevectorofasurfacecurveisindependentoftheorientationandparameterizationofthesurfaceandthecurve.However,thecurvaturevectorataparticularpointisdeterminedbythelocalshapeofthecurve,whichispartlydeterminedbythepositiononthesurfaceandthetangentialdirectiontothesurfaceatthatposition,andhencethecurvaturevectordependsonthesurfacepointandtangentialdirection,aswellasonotherfactors.ApointonthecurveatwhichthecurvaturevectorKvanishes,andhenceκ = 0,iscalledinflectionpoint.Atsuchapoint,theradiusofcurvatureisinfiniteandtheprincipalnormalvectorNandtheosculatingcirclearenotdefined.However,sinceitisusuallyassumedthatthecurveisofclassC2,thecurvaturevectorvariessmoothlyandhenceatisolatedpointsofinflectiononsuchacurve,Nmaybedefinedinsuchawaytoensurecontinuitywhenthisispossible,whichisnotalwaysthecase.OnintroducinganewunitvectorwhichisorthogonaltobothnandTanddefinedbythefollowingcrossproduct:(306)u = n × Tthecurvaturevector,whichliesintheplanespannedbynandu,canthenberesolvedinthenandudirections,whichrepresentthenormalandtangentialdirectionstothesurfaceatthegivenpoint,as:(307)K = Kn + Kg = κnn + κguwhereKnandKgarethenormalandgeodesiccomponentsofthecurvaturevectorK,whileκnandκgarethenormalandgeodesiccurvaturesofthecurveatthegivenpointrespectively.OndotproductingbothsidesofEq.307↑withnanduinturn,notingthatnanduareorthogonalunitvectors,thenormalandgeodesiccurvaturescanbeobtained,thatis:(308)κn = n⋅K =  − T⋅(dn ⁄ ds) =  − (dr ⁄ ds)⋅(dn ⁄ ds)(309)κg = u⋅K = u⋅(dT ⁄ ds) = (n × T)⋅(dT ⁄ ds)ThesecondequalityofEq.308↑isbasedonthefactthatTandnareorthogonal(sinceTistangenttothesurfacewhilenisnormaltothesurface);thereforebytheproductruleofdifferentiationwehave:(310)d(n⋅T) ⁄ ds = 

d(0) ⁄ ds = 0 = n⋅(dT ⁄ ds) + (dn ⁄ ds)⋅T = n⋅K + T⋅(dn ⁄ ds)andhence:n⋅K =  − T⋅(dn ⁄ ds).TheotherequalitiesinEqs.308↑and309↑arebasedondefinitionswhichhavebeengivenpreviously.Thevectoru,whichisaunitvectornormaltothecurveC,iscalledthegeodesicnormalvector.ThisvectoristhenormalizedprojectionofKontothetangentspaceofthesurfaceandhenceitiscontainedinthetangentplaneofthesurfaceatthegivenpoint.Also,becausethevectoruisorthogonaltothecurveCatP(sinceuisorthogonaltoTbythecrossproduct),itiscontainedinthenormalplaneofCatP.Hence,uoccursattheintersectionofthetangentplaneofthesurfaceandthenormalplaneofthecurvewhichcorrespondtothegivenpoint.Whilethenormalcurvatureκnisanextrinsicproperty,sinceitdependsonthefirstandsecondfundamentalformcoefficients,asseenin§3.6↑(Eq.260↑)andaswillbeseenin4.2↓,thegeodesiccurvatureκginanintrinsicpropertyasitdependsonlyonthefirstfundamentalformcoefficientsandtheirderivatives(see§4.3↓).Wenotethatthetriad(n, T, u)isanothermovingframewhichisincommonuseindifferentialgeometryinadditiontothecurve-basedFrenetframe(T, N, B)andthesurface-basedframe(E1,  E2,  n).RegardingtherelationbetweentheprincipalnormalvectorofthecurveN,andtheunitnormalvectortothesurfacen,letCbeacurveonasufficientlysmoothsurfaceS.IfφistheanglebetweenthevectorNofCatagivenpointPandthevectornofSatPthenwehave:(311)cosφ = n⋅NAccordingly,thenormalandgeodesiccurvatures,κnandκg,ofCatParegivenby:(312)κn = κcosφ(313)κg = κsinφwhereκisthecurvatureofCatPasdefinedpreviously(see§2.2↑andthepreviouspartsofthepresentsection).Infact,Eq.312↑canbeeasilyobtainedbycombiningEq.308↑withEq.305↑,whileEq.313↑canbesimilarlyobtainedbycombiningEq.309↑withEq.305↑,thatis:(314)κn = n⋅K = n⋅(κN) = κ(n⋅N) = κcosφ(315)κg = u⋅K = u⋅(κN) = κ(u⋅N) = κcos[(π ⁄ 2) − φ] = κsinφAccordingtothetheoremofMeusnier,ifPisagivenpointonasufficientlysmoothsurfaceS,thenallcurvesonSthatpassthroughPwiththesametangentdirectionatPhavethesamenormalcurvatureatP.ThetheoremofMeusniermayalsobestatedinthiscontextas:thecurvatureofanysurfacecurveatagivenpointPonthecurveisequalinmagnitudetothecurvatureofthenormalsectionwhichistangenttothecurveatPdividedbythecosineoftheanglebetweentheprincipalnormalvectortothecurveatPandthenormalvectortothesurfaceatP.ThisversionofthetheoremisbasedonEq.312↑plusthefactthatthenormalcurvatureκnofanormalsectionisequalinmagnitudetoitscurvatureκ.Also,weshouldexcludefromthisversionthecaseofhavingcosφ = 0whenthecurvaturevectorofthecurveistangenttothesurface.MoredetailsaboutMeusniertheoremwillbegivenin§4.2.1↓.

4.2 NormalCurvatureUsingthefirstandsecondfundamentalforms,givenbyEqs.233↑and244↑,thenormalcurvatureatagivenpointonthesurfaceinthe(du2 ⁄ du1)directioncanbeexpressedasthefollowingquotientofthesecondfundamentalforminvolvingthecoefficientsofthesurfacecovariantcurvaturetensortothefirstfundamentalforminvolvingthecoefficientsofthesurfacecovariantmetrictensor(seeEq.259↑):(316)κn = IIS ⁄ IS = [e(du1)2 + 2fdu1du2 + g(du2)2] ⁄ [E(du1)2 + 2Fdu1du2 + G(du2)2] = (bαβduαduβ) ⁄ (aγδduγduδ)wherethesymbolsareasexplainedbefore,andthelastpartistobeinterpretedasthesumofthetermsinthenumeratordividedbythesumofthetermsinthedenominator.Thisequationcanbeobtainedasfollows:(317)κn =  − T⋅(dn ⁄ ds)(Eq.308↑) =  − [(∂r ⁄ ∂uα)(duα ⁄ ds)]⋅[(∂n ⁄ ∂uβ)(duβ ⁄ ds)](Eq.106↑) =  − [(∂r ⁄ ∂uα)⋅(∂n ⁄ ∂uβ)](duα ⁄ ds)(duβ ⁄ ds) =  − {[(∂r ⁄ ∂uα)⋅(∂n ⁄ ∂uβ)]duαduβ} ⁄ (dsds) = (bαβduαduβ) ⁄ (ds)2

(Eq.213↑) = (bαβduαduβ) ⁄ (aγδduγduδ)(Eq.233↑).FromthepreviousstatementsplusEq.307↑,itcanbeseenthatthenormalcomponentofthecurvaturevectormayalsobegivenby:(318)Kn = [e(du1 ⁄ ds)2 + 2f(du1 ⁄ ds)(du2 ⁄ ds) + g(du2 ⁄ ds)2] nAlso,fromEq.316↑itcanbeseenthatthesignofthenormalcurvatureκn(i.e.beinggreaterthan,lessthanorequaltozero)isdeterminedsolelybythesignofthesecondfundamentalformsincethefirstfundamentalformispositivedefinite.Asseenbefore(see§3.6↑),thesignofthesecondfundamentalformdependsonthesurfaceorientationwhichisdeterminedbythedirectionofn.Therefore,thesignofκndependsonthesurfaceorientation.Apartfromthis,thesignofthesecondfundamentalformisrelatedtothesignofthedeterminantofthesurfacecovariantcurvaturetensorb,andhencethesignofκnisalsorelatedtothesignofb.Asgivenearlier,allsurfacecurvespassingthroughagivenpointPonasurfaceandhavethesametangentlineatPhaveidenticalnormalcurvatureatP.Hence,thenormalcurvatureisapropertyofthesurfaceatagivenpointandinagivendirectionandnotonlyapropertyofthecurve.ThenormalcurvatureκnofagivennormalsectionCofasurfaceataparticularpointPisequalinmagnitudetothecurvatureκofCatP,i.e.|κn| = κ.ThiscanbeexplainedbythefactthatthenormalvectorntothesurfaceatPiscollinearwiththeprincipalnormalvectorNofCatPsothereisonlyanormalcomponenttothecurvaturevectorwithnotangentialgeodesiccomponent.Thenormalcurvatureofasurfaceatagivenpointandinagivenspatially-fixedtangentialdirectionisaninvariantpropertywithrespecttochangeofparameterizationandrepresentationofthesurfaceapartfromitssignwhichisdependentonthechoiceofthedirectionoftheunitnormalvectorntothesurface,asseenearlier.Thenormalcurvatureisanextrinsicproperty,sinceitnecessarilydependsonthecoefficientsofthesecondfundamentalform,andhenceitcannotbeexpressedpurelyintermsofthecoefficientsofthefirstfundamentalform.Atflatpointsonasurface,κn = 0inalldirections.Atellipticpoints,κn ≠ 0inanydirectionandithasthesamesigninalldirections.Atparabolicpoints,κnhasthesamesigninalldirectionsexceptthedirectioninwhichthesecondfundamentalformvanisheswhereκn = 0.Athyperbolicpoints,κnisnegative,positiveandzerodependingonthedirection.Forthedefinitionofflat,elliptic,parabolicandhyperbolicpointsandtheirsignificance,thereaderisreferredto§4.9↓.InanytwoorthogonaltangentialdirectionsatagivenpointPonasufficientlysmoothsurface,

thesumofthenormalcurvaturescorrespondingtothesedirectionsatPisconstant.AtanypointPofasufficientlysmoothsurfaceS,thereexistsaparaboloid(ellipticorhyperbolic)whichistangentatitsvertextothetangentplaneofSatPsuchthatthenormalcurvatureoftheparaboloidinagivendirectionatPisequaltothenormalcurvatureofSatPinthatdirection.Thisparaboloidtakesthedegenerateformofaplaneatflatpointsandaparaboliccylinderatparabolicpoints(see§4.9↓).ThenormalcurvaturesofthesurfacecurvesatagivenpointPinthedirectionsoftheuandvcoordinatecurvesaregivenrespectivelyby:(319)κnu = b11 ⁄ a11 = e ⁄ E(320)κnv = b22 ⁄ a22 = g ⁄ Gwhereκnuandκnvarethenormalcurvaturesinthedirectionsoftheuandvcoordinatecurves,theindexedaandbarethecoefficientsofthecovariantmetrictensorandthecovariantcurvaturetensorandwheretheseareevaluatedatP.Thiscanbeseen,forexample,fromEq.316↑wherethelasttwotermsinthesumswillvanishfortheu1coordinatecurvesincedu2 = 0whilethefirsttwotermsinthesumswillvanishfortheu2coordinatecurvesincedu1 = 0.Wenotethatdu1 ≠ 0ontheu1coordinatecurveanddu2 ≠ 0ontheu2

coordinatecurve.Aswewillseein§4.4↓,ateachnon-umbilicalpointP(referto§4.10↓)ofasufficientlysmoothsurfacetherearetwoperpendiculardirectionsalongwhichthenormalcurvatureofthesurfaceatPtakesitsmaximumandminimumvaluesofallthenormalcurvaturevaluesatP.ThenecessaryandsufficientconditionforagivenpointPonasufficientlysmoothsurfaceStobeumbilicalpointisthatthecoefficientsofthefirstandsecondfundamentalformsofthesurfaceatPareproportional,thatis:(321)e ⁄ E = f ⁄ F = g ⁄ G( = κn)whereE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalformsatP,andκnisthenormalcurvatureofSatPinanydirection.ThiscanbeseenfromEq.316↑whereκninthiscasebecomesindependentofthedirectionsincebytakingoutthecommonproportionalityfactortheexpressionwillbereducedto:κn = cwherecistheproportionalityfactor.Moreexplicitly,accordingtoEq.321↑:e = cE,f = cFandg = cGwherecisthecommonproportionalityfactor,andhencefromEq.316↑weobtain:κn = c × 1 = c.Asseenbefore,thecurvatureκandthenormalcurvatureκnofasurfacecurveatagivenpointPonthecurvearerelatedby:(322)κn = κcosφwhereφistheanglebetweentheprincipalnormalvectorNofthecurveatPandtheunitnormalvectornofthesurfaceatP.Ateverypointonasphereandinanydirection,thenormalcurvatureisconstantgivenby:|κn| = (1 ⁄ R)whereRisthesphereradius.AtanypointPonasphere,anysurfacecurveCpassingthroughPinanydirectionisanormalsectioniffCisagreatcircle.Allthesegreatcircleshaveconstantcurvatureκandnormalcurvatureκnwhicharebothequalinmagnitudeto(1 ⁄ R)whereRisthesphereradius.Weremarkthatthegreatcirclesofaspherearetheplanesectionsformedbytheintersectionofthespherewiththeplanespassingthroughthecenterofthesphere.

4.2.1 MeusnierTheoremAccordingtothetheoremofMeusnier,allsurfacecurvespassingthroughagivenpointPonasurfaceandhavethesametangentialnon-asymptoticdirection(see§5.9↓)atPhaveidenticalnormalcurvaturewhichisthenormalcurvatureκn(andthecurvatureκconsideringthemagnitude)ofthenormalsectionatPinthegivendirection.Moreover,theosculatingcirclesofthesecurveslieonasphereSswithradius(1 ⁄ κ)andwithcenteratrC = rP + (N ⁄ κ)whereκ(whichisequalinmagnitudetoκn)isthecurvatureofthenormalsectionatP,NistheprincipalnormalvectorofthenormalsectionatPandrPisthepositionvectorofP.Asaconsequenceofthistheorem,wehave:1.ThecenterofthesphereSsisthecenterofcurvature(see§2.6↑)ofthenormalsectionatPinthegivendirection.2.ThesecurvesarecharacterizedbybeingtangenttothenormalsectionatPinthegivendirectionandbybeingplanesectionsofthesurfacewithsharedtangentdirectionatP(seeFootnote16in§8↓).3.TheosculatingcirclesofthesecurvesaretheintersectionofthesphereSswiththeosculatingplanesofthesecurvesatP.

4.ThesphereSsistangenttothetangentplaneofthesurfaceatP.ThetheoremofMeusniermayalsobestatedasfollows:thecenterofcurvatureofasurfacecurveatagivenpointPonthecurveisobtainedbyorthogonalprojectionofthecenterofcurvatureofthenormalsection,whichistangenttothecurveatP,ontheosculatingplaneofthecurve.

4.3 GeodesicCurvatureAsdescribedearlier(see§4.1↑),thecurvaturevectorKofasurfacecurveliesinaplaneperpendiculartothetangentvectorTanditcanberesolvedintoanormalcomponentKn = κnnandageodesiccomponentKg = κguwherethenormalandgeodesiccurvatures,κnandκg,aregivenbyEqs.308↑and309↑.ThegeodesiccomponentKgofthecurvaturevectorKofasurfacecurveatagivenpointPandinagivendirectionistheprojectionofKontothetangentspaceTPSofthesurfaceatP.Thisgeodesiccomponentofthecurvaturevectorofasurfacecurveisgivenby:(323)Kg = κgu = [(d2u1 ⁄ ds2) + Γ1αβ(duα ⁄ ds)(duβ ⁄ ds)]E1 + [(d2u2 ⁄ ds2) + Γ2αβ(duα ⁄ ds)(duβ ⁄ ds)]E2wheretheChristoffelsymbolsarederivedfromthesurfacemetric.Sinceu = n × T,thesenseofthegeodesiccurvaturevectordependsontheorientationofthesurfaceandtheorientationofthecurve.Thegeodesiccomponentofthecurvaturevectormayalsobegivenbythefollowingexpression:(324)

wherethesymbolsareasexplainedbefore.Aswellasthepreviouslydevelopedexpressionsforκg(seeEqs.309↑and313↑),itcanbeshownthatforanaturallyparameterizedcurvethegeodesiccurvatureκgisalsogivenby:(325)

wheretheChristoffelsymbolsarederivedfromthesurfacemetricanda = EG − F2isthedeterminantofthesurfacecovariantmetrictensor.Whilethecurvatureκandthenormalcurvatureκnareextrinsicpropertiesofthesurface,thegeodesiccurvatureκgisanintrinsicproperty.Thiscanbeseen,forexample,fromEq.325↑.Ontheu1coordinatecurves,du2 ⁄ ds = 0anddu1 ⁄ ds = 1 ⁄ √(E)(seeFootnote17in§8↓).Hence,Eq.325↑willsimplifyto:(326)κgu = √(a)Γ211(du1 ⁄ ds)3 = [√(a) ⁄ E3 ⁄ 2]Γ211whereκguisthegeodesiccurvatureoftheu1coordinatecurve.Thelastformulawillbesimplifiedfurtheriftheu1andu2coordinatecurvesareorthogonal,sinceinthiscaseF = 0andΓ211 =  − Ev ⁄ (2G)(seeEq.74↑),andtheformulawillbecome:(327)κgu =  − Ev ⁄ [2E√(G)]Similarly,ontheu2coordinatecurves,du1 ⁄ ds = 0anddu2 ⁄ ds = 1 ⁄ √(G)(seeFootnote18in§8↓).Hence,Eq.325↑willsimplifyto:(328)κgv =  − √(a)Γ122(du2 ⁄ ds)3 =  − [√(a) ⁄ G3 ⁄ 2]Γ122whereκgvisthegeodesiccurvatureoftheu2coordinatecurve.Thelastformulawillbesimplifiedfurtheriftheu1andu2coordinatecurvesareorthogonal,sinceinthiscaseF = 0andΓ122 =  − Gu ⁄ (2E)(seeEq.77↑),andtheformulathenbecomes:(329)κgv = Gu ⁄ [2G√(E)]Althoughthegeodesiccurvatureisanintrinsicproperty,ascanbeseenfromtheaboveequations,itcanalsobecalculatedextrinsicallyby:(330)κg = [r⋅(n × ṙ)] ⁄ (ṙ⋅ṙ)wheretheoverdotsstandfordifferentiationwithrespecttoageneralparametertofthecurve.Asdiscussedpreviously,someintrinsicpropertiescan

alsobedefinedintermsofextrinsicparameters.Asseenbefore,thecurvatureκandthegeodesiccurvatureκgofasurfacecurveatagivenpointPonthecurvearerelatedby:(331)κg = κsinφwhereφistheanglebetweentheprincipalnormalvectorNofthecurveatPandtheunitnormalvectornofthesurfaceatP.Itshouldberemarkedthatthegeodesiccurvaturecantakeanyrealvalue:positive,negativeorzero.However,therearesomedetailsrelatedtothedefinitionoftheangleφinthelastequationandthegeodesicnormalvectoruthatshouldbeconsidered.OnasurfacepatchofclassC2withorthogonalcoordinatecurves,ans-parameterizedcurveCofclassC2hasageodesiccurvaturegivenby:(332)κg = (dθ ⁄ ds) + κgucosθ + κgvsinθwhereκguandκgvarethegeodesiccurvaturesoftheuandvcoordinatecurvesandθistheanglesuchthat:(333)T = (E1 ⁄ | E1|)cosθ + (E2 ⁄ | E2|)sinθwhereTisthetangentunitvectorofC,andE1andE2arethesurfacebasisvectors,andwhereallthegivenquantitiesareevaluatedatagivenpointonthecurve.WenotethatEq.332↑isknownasLiouvilleformula.

4.4 PrincipalCurvaturesandDirectionsOnrotatingtheplanecontainingn(i.e.theunitnormalvectortothesurfaceatagivenpointPonthesurface)aroundn,thenormalsectionandhenceitscurvatureκandnormalcurvatureκnatPwillvaryingeneral(seeFootnote19in§8↓).Basedonthepreviousfindings(see§4.2↑),thenormalcurvatureκn(which,foranormalsection,isequalinmagnitudetoitscurvatureκ)ofthesurfaceatPinagivendirectionλcanbegivenby:(334)κn = (e + 2fλ + gλ2) ⁄ (E + 2Fλ + Gλ2)whereE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalformsandλ = du2 ⁄ du1.Infact,Eq.334↑isavariantofEq.316↑obtainedbydividingthenumeratoranddenominatorofEq.316↑by(du1)2.Thedirectionsrepresentedby(du2 ⁄ du1)arethedirectionsofthetangentstothenormalsectionsatP.Wenotethatforeaseofnotationandexpression,symbolslike(du2 ⁄ du1)anddu:dvarelaxlylabeledasdirectionssinceapairlike(du1, du2)or(du, dv)representsadirection.ThetwoprincipalcurvaturesofthesurfaceatP,κ1andκ2,whichrepresentrespectivelythemaximumandminimumvaluesofthenormalcurvatureκnofthesurfaceatPasgivenbyEq.334↑,correspondtothetwoλrootsofthefollowingquadraticequation:(335)(gF − fG)λ2 + (gE − eG)λ + (fE − eF) = 0where(gF − fG) ≠ 0.Thelastequationisobtainedbyequatingthederivativeofκn(asgivenbyEq.334↑)withrespecttoλtozerotoobtaintheextremumvalues.Eq.335↑possessestworoots,λ1andλ2,whichaccordingtotherulesofpolynomialsarelinkedbythefollowingrelations:(336)λ1 + λ2 = (gE − eG) ⁄ (gF − fG)λ1λ2 = (fE − eF) ⁄ (gF − fG)where(gF − fG) ≠ 0.Theserootsrepresentthetwodirectionscorrespondingtothetwoprincipalcurvatures,κ1andκ2,ofthesurfaceatthegivenpoint,asindicatedabove.Thefollowingtwovectorsonthesurface,whicharedefinedintermsofthetwoλrootsoftheabovequadraticequation,definethespatialdirectionscorrespondingtotheprincipalcurvatures:(337)(dr ⁄ du1)1 = (∂r ⁄ ∂u1) + λ1(∂ r ⁄ ∂u2) = E1 + λ1 E2(338)(dr ⁄ du1)2 = (∂r ⁄ ∂u1) + λ2(∂ r ⁄ ∂u2) = E1 + λ2 E2Thesedirections,whicharecalledtheprincipaldirectionsorthecurvaturedirectionsofthesurfaceatpointP,areorthogonalatnon-umbilicalpointswhereκ1 ≠ κ2.Atumbilicalpoints(see§4.10↓),thenormalcurvatureisthesameinalldirectionsandhencetherearenoprincipaldirectionstobeorthogonaloreverydirectionisaprincipaldirectionandhencethereisnosensiblemeaningforbeingorthogonal.Consequently,atanypointonaplanesurfacealldirectionsareprincipaldirectionsor,alternatively,thereisnoprincipaldirection(dependingonallowingmorethantwoprincipaldirectionsornot).Similarly,atanypointonaspherealldirectionsareprincipaldirectionsorthereisnoprincipaldirection.Itisnoteworthythattheprincipaldirectionsareinvariantwithrespecttopermissiblechangesinsurfacerepresentationandparameterization.However,weremarkthatalthoughtheprincipaldirectionsinagivencoordinatesystemoftheambientspacearefixedandhencetheirpositionandorientationrelativetothesurfaceareinvariant,theirpositionandorientationwithrespecttothesurfacecoordinatesdependonthecoordinatesystememployedtorepresentthesurface.Thepositionsofthecentersofcurvature(see§2.6↑)ofthenormalsectionscorrespondingtothetwoprincipalcurvaturesatagivenpointPonasurfaceSaregivenintensornotationby:(339)xi1 = xiP + (Ni

1 ⁄ |κ1|)(340)xi2 = xiP + (Ni

2 ⁄ |κ2|)wherexi1andxi2arethespatialcoordinatesofthefirstandsecondcenterofcurvaturecorrespondingtothetwoprincipalcurvatures,xiParethespatialcoordinatesofP,Ni

1andNi2aretheprincipalnormalvectorsofthetwonormalsectionscorrespondingtothetwoprincipalcurvatures,κ1andκ2are

theprincipalcurvaturesofSatP,andi = 1, 2, 3.WenotethattheprincipalnormalvectorNofanormalsectionatagivenpointPonthesurfaceiscollinearwiththeunitnormalvectorntothesurfaceatPandhencetheprincipalnormalvectorsareinthesameorientationforallnormalsectionsatP.However,thetwoprincipalnormalvectorscorrespondingtothetwoprincipalcurvaturesmaybeparalleloranti-parallelandhencewelabeledthemdifferentlytobegeneral(seeFootnote20in§8↓).WealsoremarkthatthetwonormalsectionsintheprincipaldirectionsatParecalledtheprincipalnormalsectionsofthesurfaceatP,whilethecentersofcurvatureoftheseprincipalnormalsectionsaredescribedastheprincipalcentersofcurvatureofthesurfaceatP.AccordingtooneoftheEulertheorems,thenormalcurvatureκnatagivenpointPonasurfaceofclassC2inagivendirectioncanbeexpressedasacombinationoftheprincipalcurvatures,κ1andκ2,atPas:(341)κn = κ1cos2θ + κ2sin2θwhereθistheanglebetweentheprincipaldirectionofκ1atPandthegivendirection.Sincetheprincipaldirectionsatnon-umbilicalpointsareorthogonal,θcouldrepresenttheanglewiththeotherprincipaldirectionbutwithrelabelingofthetwokappas.ThereareanumberofinvariantparametersofthesurfaceatagivenpointPonthesurfacewhicharedefinedintermsoftheprincipalcurvaturesatP;theseinclude:1.Theprincipalradiiofcurvature:R1 = |1 ⁄ κ1|andR2 = |1 ⁄ κ2|.2.TheGaussiancurvature:K = κ1κ2.3.Themeancurvature:H = (κ1 + κ2) ⁄ 2.Table1↓showstherestrictingconditionsontheprincipalcurvatures,κ1andκ2,foranumberofcommonsurfaceswithsimplegeometricshapes(plane,cylinder,sphere,ellipsoidandhyperboloidofonesheet)andtheeffectontheGaussiancurvatureKandthemeancurvatureH.Table1 Thelimitingconditionsontheprincipalcurvatures,κ1andκ2,foranumberofsurfacesofsimplegeometricshapes(“Hyp.1S”standsforhyperboloidofonesheet)alongsidethecorrespondingmeancurvatureHandGaussiancurvatureK.Apartfromtheplane,theunitnormalvectortothesurface,n,isassumedtobeintheoutsidedirection.__________________

κ1 κ2 H KPlane 0 0 0 0Cylinder κ1 = 0 κ2 < 0 H < 0 0Sphere κ1 = κ2 < 0 κ2 = κ1 < 0 H < 0 K > 0Ellipsoid κ1 < 0 κ2 < 0 H < 0 K > 0Hyp.1S κ1 > 0 κ2 < 0 --- K < 0TheGaussiancurvaturemayalsobecalledthe“Riemanniancurvature”.However,the“Riemanniancurvature”isusuallyusedtolabelthistypeofcurvatureforgeneralnDspaceswhilethe“Gaussiancurvature”isbeingusedtolabelthespecialinstanceofitthatappliesto2Dspaces,andhencetheGaussiancurvatureistheRiemanniancurvatureofsurfaces.Itshouldberemarkedthatsomeauthorsuse“totalcurvature”forthe“Gaussiancurvature”andhencethesetwotermsaresynonym,whileothersuse“totalcurvature”fortheareaintegral∬Kdσasused,forexample,intheGauss-Bonnettheorem(referto§4.8↓).Inthepresentbook,weusetotalcurvaturestrictlyfortheintegralandhencewelabeltheGaussiancurvaturewithKandthetotalcurvaturewithKt.AnotherremarkisthatsomeauthorsdefineHasthesumofκ1andκ2,thatis:H = κ1 + κ2,ratherthantheaverageasdefinedabove.Infacteachoneoftheseconventionshasitsmerit.However,inthepresentbookwedefineHastheaverage,notthesum,ofthetwoprincipalcurvatures.Intheneighborhoodofagivenpointonasurface,thesurfacecanbeapproximatedbyaquadraticexpressioninvolvingtheprincipalcurvaturesatthatpoint.Moreformally,letPbeapointonasufficientlysmoothsurfaceSembeddedina3DspacecoordinatedbyarectangularCartesiansystem(x, y, z)withPbeingabovetheorigin,thetangentplaneofS(x, y)atPbeingparalleltothexyplane,andtheprincipaldirectionsbeingalongthexandycoordinatelines.TheequationofSintheneighborhoodofPcanthenbeexpressed,upandincludingthequadraticterms,inthefollowingform:(342)S(x, y)≃S(0, 0) + [(κ1x2) ⁄ 2] + [(κ2y2) ⁄ 2]

whereκ1andκ2aretheprincipalcurvaturesofSatP.ThismeansthatintheimmediateneighborhoodofP,Sresemblesaquadraticsurface(see§6.2↓)ofthegivenform.Theaboveformincludesumbilicalpoints(see§4.10↓)wheretheprincipaldirectionscanbearbitrarilychosenasthedirectionsofthexandycoordinatelines.Thenecessaryandsufficientconditionforanumberκ ∈ ℝtobeaprincipalcurvatureofasmoothsurfaceSatagivenpointPandinagivendirection(dv ⁄ du),where(du)2 + (dv)2 ≠ 0,isthatthefollowingequationsaresatisfied:(343)(e − κE)du + (f − κF)dv = 0(344)(f − κF)du + (g − κG)dv = 0whereE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalformsatP.Itisworthnotingthatforsimplicityinnotation,weuseκintheseandthefollowingequationstorepresentprincipalcurvature.Thisuseshouldnotbeconfusedwiththecurvecurvaturewhichisalsosymbolizedbyκ.However,forthiscasethecurvatureisequal(inmagnitudeatleast)totheprincipalcurvaturesincethelatteristhecurvatureofanormalsectionandhencetheuseofκisjustified.Theaboveequationscanbecastinamatrixformas:(345)

Thissystemofhomogeneouslinearequationshasanon-trivialsolution(du, dv)iffthedeterminantofthecoefficientmatrixiszero,thatis:(346)

(EG − F2)κ2 − (gE − 2fF + eG)κ + (eg − f2) = 0Basedonthegivenconditions,thisquadraticequationinκhasanon-negativediscriminantandhenceitpossesseseithertwodistinctrealrootsorarepeatedrealroot.IntheformercasetherearetwodistinctprincipalcurvaturesatPcorrespondingtotwoorthogonalprincipaldirections,whileinthelattercasethepointisumbilicalwhereallthenormalsectionsatthepointhavethesame“principalcurvature”althoughthereisnospecificprincipaldirectionsinceeachdirectioncanbeaprincipaldirection.Soinbrief,agivenrealnumberκisaprincipalcurvatureofSatPiffitisasolutionofEq.346↑.FromEq.346↑,itcanbeseenthattheprincipalcurvaturesofasurfaceatagivenpointParethesolutionsofthisquadraticequationandhencetheyaregivenby:(347)

OndividingEq.346↑bya = EG − F2(whichispositivedefiniteasestablishedbefore)weobtain:(348)κ2 − 2Hκ + K = 0whereHandKarethemeanandGaussiancurvatureswhoseexpressionsintermsofthecoefficientsofthefirstandsecondfundamentalformsaretakenfromEqs.383↓and356↓.Hence,Eq.347↑canbeexpressedcompactlyas:(349)κ1, 2 = H±√(H2 − K)Infact,thisformulacanbeobtaineddirectlyfromEq.348↑usingthequadraticformula.Theaboveconditionsabouttheprincipalcurvaturesmaybestatedratherdifferentlyintermsoftheprincipaldirectionsthatis,foranon-umbilicalpointPonasufficientlysmoothsurfaceS,adirection(du2 ⁄ du1)isaprincipaldirectionofSatPiffthefollowingconditionistrue:(350)(fE − eF)du1du1 + (gE − eG)du1du2 + (gF − fG)du2du2 = 0Thelastequation,whichisobtainedfromEq.335↑bymultiplyingbothsideswith(du1)2,canbefactoredintotwolinearequationseachoftheform:Adu1 + Bdu2 = 0

(withAandBbeingrealconstants)wheretheseequationsrepresentthetwoorthogonalprincipaldirections.Similarly,atagivennon-umbilicalpointPonasufficientlysmoothsurfaceS,adirection(dv ⁄ du)isaprincipaldirectioniffforarealnumberκthefollowingrelationholdstrue:(351)dn =  − κdrwhere:(352)dn = (∂n ⁄ ∂u)du + (∂n ⁄ ∂v)dvdr = (∂r ⁄ ∂u)du + (∂r ⁄ ∂v)dvIfthisconditionissatisfied,thenκistheprincipalcurvatureofSatPcorrespondingtotheprincipaldirection(dv ⁄ du).Eq.351↑isknownastheRodriguescurvatureformula.TheobviousinterpretationoftheRodriguesformulaisthatinanyprincipaldirectionthetwovectorsdnanddrhavethesameorientationwheretheprincipalcurvatureκinthatdirectionisthescalefactorbetweenthetwovectors.FromtheRodriguescurvatureformula,thefollowingsubsidiaryequationscorrespondingtothesurfacecoordinatecurvescanbeeasilyobtained:(353)∂n ⁄ ∂u =  − κE1∂n ⁄ ∂v =  − κE2Oneachnon-umbilicalpointPofasmoothsurfaceSanorthonormalmoving“Darbouxframe”canbedefined.Thisframeconsistsofthevectortriad(d1,  d2,  n)whered1andd2aretheunitvectorscorrespondingtotheprincipaldirectionsatP,andn = d1 ×  d2istheunitnormalvectortothesurfaceatP.Thisisanothermovingframeinuseindifferentialgeometryinadditiontothethreepreviously-describedframes:the(T,N,B)frame,the(E1,  E2,  n)frameandthe(n, T, u)frame(see§1.4.5↑,2.5↑,3.2↑and4.1↑).Thefirstoftheseframes,i.e.(T,N,B),isassociatedwithcurveswhiletheremainingthreeareassociatedwithsurfaces.Whatiscommontoallthesefourframesisthattheyaremovingframeswhosevectorscanbeusedasbasissetsfortheembedding3Dspacesinceeachoneofthesesetsconsistsofthreelinearlyindependentvectors.Also,allthesesets,except(E1,  E2,  n),areorthonormal.WhentheuandvcoordinatecurvesofasurfaceatagivenpointParealignedalongtheprincipaldirectionsatP,theprincipalcurvaturesatPwillbegivenby(see§4.2↑):(354)κ1 = b11 ⁄ a11 = e ⁄ Eκ2 = b22 ⁄ a22 = g ⁄ Gwheretheindexedaandbarethecoefficientsofthesurfacecovariantmetricandcovariantcurvaturetensors,andE, G, e, garethecoefficientsofthefirstandsecondfundamentalformsatP.ThismaybeobtainedfromEq.316↑wherethelasttwotermsinthesumswillvanishfortheucoordinatecurvesincedv = 0whilethefirsttwotermsinthesumswillvanishforthevcoordinatecurvesincedu = 0.Wenotethatweareassuminghereaparticularlabelingoftheuandvcoordinatecurvesforthelabelingofthetwokappastobeappropriate,i.e.theucoordinatecurveisalignedalongthefirstprincipaldirectionandthevcoordinatecurveisalignedalongthesecondprincipaldirection.Itshouldberemarkedthatonanorientedandsufficientlysmoothsurface,theprincipalcurvatures,κ1andκ2,arecontinuousfunctionsofthesurfacecoordinates.Anotherremarkisthattheprincipalcurvaturesaretheeigenvaluesofthemixedtypesurfacecurvaturetensorbαβ.

4.5 GaussianCurvatureTheGaussiancurvature,whichmayalsobecalledtheRiemanniancurvatureofthesurface,representsageneralizationofcurvecurvaturetosurfacessinceitistheproductoftwocurvaturesofcurvesembeddedinthesurfaceandhenceinthissenseitisa2Dcurvature.Asgivenearlier,theGaussiancurvatureKatagivenpointPonasurfaceisdefinedastheproductofthetwoprincipalcurvatures,κ1andκ2,ofthesurfaceatPthatis:(355)K ≡ κ1κ2TheGaussiancurvatureofasurfaceatagivenpointPonthesurfaceisgivenby:(356)K = (eg − f2) ⁄ (EG − F2) = b ⁄ a = R1212 ⁄ awhereE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalformsatP,aandbarethedeterminantsofthesurfacecovariantmetricandcovariantcurvaturetensors,andR1212isthecomponentofthe2DcovariantRiemann-Christoffelcurvaturetensor.FromEq.231↑wehave:(357)R1212 = b11b22 − b12b21 = eg − f2 = bwheretheindexedbarethecoefficientsofthesurfacecovariantcurvaturetensor,andhencetheequalitiesinEq.356↑arefullyjustified.Asdiscussedpreviously(see§1.4.10↑),the2DRiemann-Christoffelcurvaturetensorhasonlyoneindependentnon-vanishingcomponentwhichisrepresentedbyR1212.Therefore,Eq.356↑providesafulllinkbetweentheGaussiancurvatureandtheRiemann-Christoffelcurvaturetensor.Theaboveformulae(Eq.356↑)arealsobasedonthefactthattheGaussiancurvatureKisthedeterminantofthemixedcurvaturetensorbαβofthesurface,thatis:(358)K = det(bαβ) = det(aαγbγβ) = det(aαγ)det(bγβ) = det(bγβ) ⁄ det(aαγ) = b ⁄ awherethesymbolsareasdefinedpreviously.FromEq.356↑,weseethatthesignofK(i.e.K > 0,K < 0orK = 0)isthesameasthesignofbandthesignofR1212sinceaispositivedefinite.WenotethatbeingthedeterminantofatensorestablishesthestatusofKasaninvariantunderpermissiblecoordinatetransformations.WealsonotethatthechainofformulaeinEq.358↑maybetakenintheoppositedirectionstartingprimarilyfromK = (b ⁄ a)orK = (R1212 ⁄ a)asadefinitionorasaderivedresultfromotherarguments,andhencethestatementK = det(bαβ)willbeobtainedasasecondaryresult.SincebothR1212(seeEq.88↑)andadependexclusivelyonthesurfacemetrictensor,Eq.356↑revealsthatKdependsonlyonthefirstfundamentalformcoefficientsandhenceitisanintrinsicpropertyofthesurface(referto§4.7↓).ThedependenceofKonthesecondfundamentalformcoefficientsinEq.356↑orEq.358↑doesnotaffectitsqualificationasanintrinsicpropertysincethisdependencyisnotindispensableasKcanbeexpressedintermsofthefirstfundamentalformcoefficientsexclusively.Infact,accordingtoEq.262↑evenbcanbeexpressedexclusivelyintermsofthefirstfundamentalformcoefficients.BecausetheGaussiancurvatureisaninvariantwithrespecttopermissiblecoordinatetransformationsin2Dmanifolds,wehave:(359)K = R1212 ⁄ a = R1212 ⁄ ãwherethebarredandunbarredsymbolsrepresentthequantitiesinthebarredandunbarredcoordinatesystems.TheGaussiancurvatureisalsoinvariantwithrespecttothetypeofrepresentationandparameterizationofthesurface.Inparticular,theGaussiancurvatureisindependent,insignandmagnitude,oftheorientationofthesurfacewhichisbasedonthechoiceofthedirectionofthenormalvectorntothesurface.Thisisbecauseachangeinthedirectionofnwillchangethesignoftheprincipalcurvaturesbutnottheirabsolutevalueandhencethemagnitudeispreserved.Furthermore,thischangeofsignwillnotaffectthesignoftheGaussiancurvaturesincebothsignswillbechangedbythereversalofndirectionandhencetheirproductwillnotbeaffected.Therefore,thesignandmagnitudeoftheGaussiancurvaturearebothpreservedunderthisreversal.FromTable1↑weseethattheGaussiancurvatureofplanesandcylindersarebothidenticallyzero.AttherootofthisisthefactthattheGaussiancurvatureisanintrinsicpropertyandthecylinderisadevelopablesurfaceobtainedbywrappingaplanewithnolocalizeddistortionbystretchingorcompression.Hence,theplanesandcylinderspossessidenticalfirstfundamentalforms,asindicatedpreviouslyin§3.5↑,andconsequentlytheyhaveidenticalGaussiancurvature(alsosee§4.7↓).SincethemagnitudeofthenormalcurvatureofasphereofradiusRis|κn| = (1 ⁄ R)atanypointonitssurfaceandforanynormalsectioninanydirection,itsGaussiancurvatureisaconstantgivenbyK = (1 ⁄ R2).ForaMongepatchoftheformr(u, v) = (u, v, f(u, v)),theGaussiancurvatureisgivenby:(360)K = (fuufvv − fuv2) ⁄ (1 + fu2 + fv2)2wherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates.ThelastequationcanbeobtainedbycombiningEq.356↑(orEq.358↑)withEqs.201↑and226↑.TheGaussiancurvatureofasurfaceofrevolutiongeneratedbyrevolvingaplanecurveofclassC2havingtheformy = f(x)aroundthex-axisisgivenby:(361)K =  − fxx ⁄ [f(1 + fx2)2]wherethesubscriptxrepresentsderivativeoffwithrespecttothisvariable.AtanypointonasufficientlysmoothsurfacetheGaussiancurvaturesatisfiesthefollowingrelation:(362)∂un × ∂vn = K(E1 ×  E2)Ondotproductingbothsideswithnweobtain:(363)n⋅(∂un × ∂vn) = Kn⋅(E1 ×  E2) = K√(a)wherethelastequalityisbasedonEq.169↑.Hence:(364)K = [n⋅(∂un × ∂vn)] ⁄ √(a)InthelastequationtheGaussiancurvature,whichisanintrinsicproperty,isexpressedintermsofthenormalvectorn,whichisanextrinsicentity,anditsderivativesaswellasthemetrictensor.TherearesurfaceswithconstantzeroGaussiancurvatureK = 0(e.g.planes,cylindersandconesexcludingtheapex),surfaceswithconstantpositiveGaussiancurvatureK > 0(e.g.sphereswithK = (1 ⁄ R2)whereRisthesphereradius),andsurfaceswithconstantnegativeGaussiancurvatureK < 0(e.g.Beltramipseudo-spheres,seeninFig.14↑,withK = ( − 1 ⁄ ρ2)whereρisthepseudo-radiusofthepseudo-sphere).However,ingeneraltheGaussiancurvatureisavariablefunction,insignandmagnitude,ofthesurfacecoordinatesandhenceasinglesurfacecanhaveGaussiancurvatureofdifferentsignsandmagnitudes.Itisnoteworthythatsurfaceswithconstantnon-zeroGaussiancurvatureKmaybedescribedassphericalifK > 0andpseudo-sphericalifK < 0.Onscalingasurfaceupordownbyaconstantfactorc > 0,theGaussiancurvatureKwillscalebyafactorof1 ⁄ c2.Thisisbasedonthefactthatscalingthesurfacebyaconstantfactorc > 0isequivalenttoscalingthecoefficientsofthesurfacemetrictensorbyc2(seeEq.187↑)andscalingthecoefficientsofthesurfacecurvaturetensorbyc(seeEq.214↑),andhenceaccordingtoEq.356↑orEq.358↑,Kwillbescaledbyafactorof1 ⁄ c2.Thisleadstotheconclusionthatwhenthesurfacecurvatureisanon-zeroconstant,thesurfacecanbescaledupordowntomakeitsGaussiancurvature1or − 1andhencesimplifytheformulationsandcalculations.BasedonthepreviousstatementsplusthefactthattheGaussiancurvatureisanintrinsicproperty,theGaussiancurvatureisinvariantwithrespecttoallisometrictransformationssince,intrinsically,thesetransformationscorrespondtoscalingthesurfacewithunityeventhoughtheshapeofthesurfacemayhavebeendeformedextrinsically.Hence,twoisometricsurfaceshaveidenticalGaussiancurvatureateachpairoftheircorrespondingpoints.However,twosurfaceswithequalGaussiancurvatureattheircorrespondingpointsarenotnecessarilyisometric.Yes,inthecaseoftwosufficientlysmoothsurfaceswithequalconstantGaussiancurvaturethetwosurfaceshavelocalisometry.Thedetailscanbefoundinmoreadvancedbooksondifferentialgeometry.In3Dmanifolds,thereisnocompactsurfaceofclassC2withnon-positiveGaussiancurvature(i.e.K ≤ 0)overthewholesurface.Also,anycompactsurface,excludingthesphere,shouldhavepointswithnegativeGaussiancurvature.Infact,thesphereistheonlyconnected,compactandsufficientlysmoothsurfacewithconstantGaussiancurvature.AccordingtotheHilbertlemma,ifPisapointonasufficientlysmoothsurfaceSwithκ1andκ2beingtheprincipalcurvaturesofSatPsuchthat:κ1 > κ2,κ1isalocalmaximum,andκ2isalocalminimum,thentheGaussiancurvatureofSatPisnon-positive,thatisK ≤ 0.AtagivenpointPonaspherically-mapped(see§3.10↑)andsufficientlysmoothsurfaceS,theratiooftheareaofthesphericalimageZofamappedregionZsurroundingPonStotheareaofZconvergestotheabsolutevalueoftheGaussiancurvatureatPasZshrinkstoP(seeEq.303↑).FromtheGauss-Bonnettheorem(see§4.8↓),itcanbeshownthatasurfacewillhaveidentically-vanishingGaussiancurvatureifatanypointPonthesurfacetherearetwofamiliesofgeodesiccurves(see§5.7↓)intheneighborhoodofPintersectingataconstantangle.FromEqs.93↑and356↑,itcanbeseenthattheGaussiancurvatureKofasufficientlysmoothsurfacerepresentedbyr = r(u, v) = r(u1, u2)canalsobe

givenby:(365)K = [Fuv − (Evv ⁄ 2) − (Guu ⁄ 2) + aαβ(Γα12Γβ12 − Γα11Γβ22)] ⁄ awhereE, F, Garethecoefficientsofthefirstfundamentalform,thesubscriptsuandvstandforpartialderivativeswithrespecttothesesurfacecoordinates,a = EG − F2isthedeterminantofthesurfacecovariantmetrictensor,theindexedaareitscoefficients,andα, β = 1, 2.TheChristoffelsymbolsinthelastequationarebasedonthesurfacemetric.TheGaussiancurvatureofasmoothsurfaceofclassC3representedbyr(u, v)mayalsobegivenby:(366)

wherethesymbolsareasdefinedabove.Accordingly,theGaussiancurvatureofasurfaceofclassC3representedbyr(u, v)withorthogonalsurfacecoordinatecurvesisgivenby:(367)

ThisformulaisobtainedfromthepreviousformulabysettingF = 0identicallyduetotheorthogonalityofthesurfacecoordinatecurves.Thelastformulawillsimplifyto:(368)K =  − [∂uu√(G)] ⁄ √(G)whenthesurfacer(u, v)isrepresentedbygeodesiccoordinates(see§1.4.8↑)withtheucoordinatecurvesbeinggeodesicsanduisanaturalparameter(seeFootnote21in§8↓).TheGaussiancurvatureKcanalsobeexpressedintermsofthemeancurvatureH(see§4.6↓),thatis:(369)K = (H + C)(H − C) = H2 − C2

whereCisgivenby:(370)C = √([e2G2 + E2g2] − 4fF[eG + Eg] + 4[f2EG + F2eg] − 2egEG) ⁄ [2(EG − F2)]andE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalforms.ThiscanbeverifiedbytransformingEq.369↑tothefollowingform:C2 = H2 − KandsubstitutingforHandKfromEqs.383↓and356↑.TheGaussiancurvatureofasurfaceSatagivenpointPonthesurfaceispositiveifallthesurfacepointsinadeletedneighborhoodofPonSareonthesamesideofthetangentplanetoSatP.TheGaussiancurvatureisnegativeifforalldeletedneighborhoodsofPonSsomepointsareononesideofthetangentplaneandsomeareontheotherside.TheGaussiancurvatureiszeroif,inadeletedneighborhood,eitherallthepointslieinthetangentplaneorallthepointsareononesideexceptsomewhichlieonacurveinthetangentplane.Hence:1.AspherehaspositiveGaussiancurvatureatallpoints.2.Ahyperbolicparaboloid(Fig.8↑)hasnegativeGaussiancurvatureatallpoints.Similarly,themonkeysaddle(Fig.12↑)hasnegativeGaussiancurvatureatallpointsexcepttheorigin(x, y, z) = (0, 0, 0)whichisanumbilicalpoint(see§4.10↓)withzeroGaussiancurvature.3.AplanehaszeroGaussiancurvatureatallpoints.4.AcylinderhaszeroGaussiancurvatureatallpoints.5.Atorus(Fig.38↓)haspointswithpositiveGaussiancurvature(outerhalf),pointswithzeroGaussiancurvature(topandbottomcircles)andpointswithnegativeGaussiancurvature(innerhalf).

Figure38 PointsoftoruswithpositiveGaussiancurvature(outerblue),pointswithzeroGaussiancurvature(middleyellow)andpointswithnegativeGaussiancurvature(innerred).Basedontheabovestatements,theGaussiancurvatureofadevelopablesurface(see§6.4↓)isidenticallyzero.Hence,besidetheplane,thereareothersurfaceswithconstantzeroGaussiancurvaturesuchascones,cylindersandtangentsurfacesofspacecurves(referto§6.6↓).ExamplesoftheGaussiancurvature,K,foranumberofsimplesurfacesare:1.Plane:K = 0.2.SphereofradiusR:K = 1 ⁄ R2.3.Torusparameterizedbyx = (R + rcosφ)cosθ,y = (R + rcosφ)sinθandz = rsinφ:K = cosφ ⁄ [r(R + rcosφ)].ThetotalcurvatureKtisdefinedastheareaintegraloftheGaussiancurvatureKoverasurfaceorapatchofasurface,S,thatis:(371)Kt = ∬SKdσwheredσsymbolizesinfinitesimalareaelementonthesurfaceandwhereKisafunctionofthesurfacecoordinatesingeneral.FromEqs.206↑and282↑,itcanbeseenthatthetotalcurvatureKtmaybegivenby:(372)Kt ≡ ∬SKdσ = ∬SK|E1 ×  E2|dudv = ∬Ssgn(K)|∂un × ∂vn|dudvwheresgn(K)isthesignfunctionofKasafunctionofthesurfacecoordinates,uandv.TheRiemann-ChristoffelcurvaturetensorisrelatedtotheGaussiancurvaturethroughtheabsolutepermutationtensorofthesurfacebythefollowingrelation:(373)Rαβγδ = Kεαβεγδ

wheretheindexedεarethe2Dcovariantabsolutepermutationtensorsandalltheindicesrangeover1and2.Onmultiplyingbothsidesofthelastequationbyεαβεγδweget:(374)εαβεγδRαβγδ = KεαβεγδεαβεγδNow,sinceεαβεαβ = εγδεγδ = 2,thelastequationbecomes:(375)K = (1 ⁄ 4)εαβεγδRαβγδ = (1 ⁄ 4)εαβεγδ(bαγbβδ − bαδbβγ)wheretheindexedbarethecomponentsofthesurfacecovariantcurvaturetensor,andwherethelaststepisbasedonEq.227↑.ThelastequationisademonstrationofthefactthatKisanabsoluterank-0tensorsinceitisrepresentedinbothequalitiesbyacombinationofabsolutetensorswithalltheindicesofthesetensorsbeingconsumedbycontraction.TheGaussiancurvatureisalsolinkedtotheRiemann-Christoffelcurvaturetensor,throughthesurfacemetrictensor,bythefollowingrelation:(376)Rαβγδ = K(aαγaβδ − aαδaβγ)Infact,Eq.356↑isaninstanceofthelastequationwithα = γ = 1andβ = δ = 2.TheothercombinationsofindexvaluesprovidethelinkbetweenKandtheotherelementsofRαβγδ.WenotethatEq.376↑maybeextendedtonDspacesforn > 2andwithK(representingRiemanniancurvature)beingconstantbutthisisoutofthescopeofthisbook.TheGaussiancurvatureKmayalsobegivenbythefollowingrelation:(377)K = (1 ⁄ 2)εαβεγδbγαbδβwheretheindexedεarethe2Dcontravariantabsolutepermutationtensors.FromEqs.227↑and373↑,itcanbeseenthattheGaussiancurvatureandthesurfacecurvaturetensorarealsorelatedby:(378)Kεαβεγδ = bαγbβδ − bαδbβγOtherformulaefortheGaussiancurvature(intermsofthesurfacebasisvectors,theirderivativesandthecoefficientsofthefirstfundamentalform)mayalsobeobtainedfromtheformulaK = (b ⁄ a)bymanipulatingbasfollows:(379)b = eg − f2 = [(∂uE1⋅ E1 ×  E2)(∂vE2⋅ E1 ×  E2) − (∂vE1⋅ E1 ×  E2)2] ⁄ a = [(∂uE1⋅ E1 ×  E2)(∂vE2⋅ E1 ×  E2) − (∂vE1⋅ E1 ×  E2)2] ⁄ (EG − F2) = [(∂uE1⋅ E1 ×  E2)(∂vE2⋅ E1 ×  E2) − (∂vE1⋅ E1 ×  E2)2] ⁄ | E1 ×  E2|2

wherethesestepsarebasedonEqs.253↑-255↑andEq.170↑aswellastheobviousfactthata = EG − F2.Hence:(380)K = b ⁄ a = [(∂uE1⋅ E1 ×  E2)(∂vE2⋅ E1 ×  E2) − (∂vE1⋅ E1 ×  E2)2] ⁄ a2 = [(∂uE1⋅ E1 ×  E2)(∂vE2⋅ E1 ×  E2) − (∂vE1⋅ E1 ×  E2)2] ⁄ (EG − F2)2 = [(∂uE1⋅ E1 ×  E2)(∂vE2⋅ E1 ×  E2) − (∂vE1⋅ E1 ×  E2)2] ⁄ | E1 ×  E2|4Finally,ona2Dsurface,theGaussiancurvatureKisrelatedtotheRiccicurvaturescalarℛ(see§1.4.11↑)bythefollowingrelation:(381)|K| = |ℛ| ⁄ 2Asseen,theGaussiancurvatureisjustaconstantmultipleoftheRiccicurvaturescalarofthesurfaceandhencetheyareessentiallythesame.Detailsaboutthesignsofthesecurvatureparametersshouldbesoughtinmoreexpandedtextbooksondifferentialgeometryandtensoranalysis.

4.6 MeanCurvatureThemeancurvatureofasurfaceatagivenpointPisameasureoftherateofchangeofareaofthesurfaceelementsintheneighborhoodofPwithrespecttothesurfacecoordinates.Asgivenearlier,themeancurvatureHisdefinedastheaverage(seeFootnote22in§8↓)ofthetwoprincipalcurvatures,κ1andκ2,thatis:(382)H ≡ (κ1 + κ2) ⁄ 2Themeancurvatureisgivenbythefollowingformula:(383)H = (eG − 2fF + gE) ⁄ [2(EG − F2)] = tr(bαβ) ⁄ 2 = bαα ⁄ 2whereE, F, G, e, f, garethecoefficientsofthefirstandsecondfundamentalforms,theindexedbrepresentthesurfacemixedcurvaturetensor,trstandsforthetraceofmatrix,andα, β = 1, 2.ThefirstequalitycanbeobtainedbycombiningEq.382↑withEq.347↑,whilethesecondequalitycanbeverifiedbytakingthetraceofbαβasgivenbyEq.223↑.Thethirdequalityisjustamatterofdifferentsymbolismaccordingtothematrixandtensornotations.UnliketheGaussiancurvature,thesignofthemeancurvatureHisdependentonthechoiceofthedirectionoftheunitnormalvectortothesurface,n.ThiscanbeseenfromEq.382↑wherethesignsofbothkappaswillbereversedbythechangeofndirectionalthoughthemagnitudeofkappas,andhencethemagnitudeofH,willnotbeaffectedbythischange.LiketheGaussiancurvature,themeancurvatureisinvariantunderpermissiblecoordinatetransformationsandrepresentationsaslongasthesurfaceorientationispreserved.BeinghalfthetraceofatensorestablishesthestatusofHasaninvariantunderpermissiblecoordinatetransformations.Examplesofthemeancurvature,H,foranumberofsimplesurfacesare:1.Plane:H = 0.2.SphereofradiusR:|H| = 1 ⁄ R.Asstatedabove,thesignofHdependsonthechoiceofndirectionbeinginwardoroutward.3.Torusparameterizedbyx = (R + rcosφ)cosθ,y = (R + rcosφ)sinθandz = rsinφ:|H| = |(R + 2rcosφ) ⁄ (2r[R + rcosφ])|.ThesignofHinthiscasedependsonthelocationofthepointonthesurfaceaswellasthechoiceofndirection.ForaMongepatchoftheformr(u, v) = (u, v, f(u, v)),themeancurvatureisgivenby:(384)H = [(1 + fv2)fuu − 2fufvfuv + (1 + fu2)fvv] ⁄ [2(1 + fu2 + fv2)3 ⁄ 2]wherethesubscriptsuandvstandforpartialderivativesoffwithrespecttothesesurfacecoordinates.ThisequationcanbeobtainedfromthefirstequalityofEq.383↑wherethecoefficientsofthefirstandsecondfundamentalformsofMongepatchareobtainedfromEqs.201↑and226↑.Themeancurvaturemaybeconsideredasthe2Dequivalentofthegeodesiccurvaturein1D.Theequivalencecanbeunderstoodinthesensethatthemeancurvatureisameasureforextremizingsurfaceareawhilethegeodesiccurvatureisameasureforextremizingcurvelength.Accordingly,the2Dminimalsurfaces(see§6.7↓)correspondtothe1Dgeodesiccurves(see§5.7↓).

4.7 TheoremaEgregiumTheessenceofGaussTheoremaEgregiumorRemarkableTheoremisthattheGaussiancurvatureKofasurfaceisanintrinsicpropertyofthesurfaceandhenceitcanbeexpressedasafunctionofthecoefficientsofthefirstfundamentalformandtheirpartialderivativesalonewithnoinvolvementofthecoefficientsofthesecondfundamentalform.ThiscanbeguessedforexamplefromthelastpartofEq.356↑.Infact,eventhefirstpartofEq.356↑canbeusedinthisargumentsincebcanbeexpressedpurelyintermsofthecoefficientsofthefirstfundamentalformandtheirderivativesaccordingtoEq.262↑.TheessenceofTheoremaEgregium,asastatementofthefactthatcertaintypesofcurvatureareintrinsictothesurface,iscontainedinseveralformsandequations;someofwhichareindicatedinthisbookwhentheyoccur.Forexample,Eq.227↑whichlinksthesurfacecurvaturetensortotheRiemann-Christoffelcurvaturetensor(whichisanintrinsicpropertyofthesurfaceandisrelatedtotheGaussiancurvaturebyEq.373↑forinstance)canberegardedasastatementofTheoremaEgregiumsinceitexpressesaformofsurfacecurvaturerepresentedbyacertaincombinationofthecoefficientsofthecurvaturetensorintermsofacombinationofpurelyintrinsicsurfaceparameters.AnexamplemaybegiventodemonstratethesignificanceofTheoremaEgregiumthatis,ifapieceofplaneisrolledintoacylinderofradiusR,thenκ1, κ2, Hwillchangefrom0, 0, 0to(1 ⁄ R), 0, (1 ⁄ [2R])where,forthecylinder,weareassuminganormalunitvectornintheinnerdirection.However,asaconsequenceofTheoremaEgregium,KwillnotchangesinceKisdependentexclusivelyonthefirstfundamentalformwhichisthesameforplanesandcylindersasstatedpreviously.AccordingtoTheoremaEgregium,theGaussiancurvatureofasufficientlysmoothsurfaceofclassC3atagivenpointPcanberepresentedbythefollowingfunctionofthecoefficientsofthefirstfundamentalformandtheirpartialderivativesatP:(385)

whereC = ( − Evv + 2Fuv − Guu) ⁄ 2andthesubscriptsuandvstandforpartialderivativeswithrespecttothesesurfacecoordinates.Theothersymbolsandnotationsareasdefinedpreviously.

4.8 Gauss-BonnetTheoremThistheoremtiesthegeometryofsurfacestotheirtopology.Thereareseveralvariantsofthistheorem;someofwhicharelocalwhileothersareglobal.Duetotheimportanceandsubtletyofthistheoremwegivetwovariantsofthetheoremandseveralexamplesfrombothplaneandtwistedsurfaces.AccordingtotheGauss-Bonnettheorem,ifDisasimplyconnectedregiononasurfaceofclassC3whereDisborderedbyafinitenumbermofpiecewiseregularcurvesCjthatmeetinncornersthenwehave:(386)

wherethefirstsumisoverthecurveswhilethesecondsumisoverthecorners,κgisthegeodesiccurvatureofthecurvesCjasafunctionoftheircoordinates,φkaretheexterioranglesofthecornersandKistheGaussiancurvatureofDasafunctionofthecoordinatesoverD.ThegeodesicandGaussiancurvaturesintheaboveformulationshouldbecontinuousandfiniteovertheirdomain.Asindicatedpreviously,theterm∬DKdσ,whichrepresentstheareaintegraloftheGaussiancurvatureovertheregionDoftheabove-describedsurface,iscalledthetotalcurvatureKtofD.Wenotethatthecornersindicatedinthelastparagraphcanbedefinedasthepointsofdiscontinuityofthetangentsoftheboundarycurves.Theanglesofthesecornersarethereforedefinedastheanglesbetweenthetangentvectorsatthepointsofdiscontinuitywhentraversingtheboundarycurvesinapredefinedsense.Asindicatedabove,theseanglesareexteriortotheregionsurroundedbythecurves.Sometimes,“artificialcorners”atregularpointsareintroducedforconveniencetoestablishanargument;inwhichcasetheexteriorangleiszero.Severalexamplesrelatedtoartificialcornerswillbegivenintheforthcomingpartsofthebook.ItshouldberemarkedthattheformoftheGauss-BonnettheoremgivenbyEq.386↑maybelabeledasalocalvariantofthetheoremalthoughitslocalitymaynotbeobvious.However,itcanbejustifiedbyitsapplicationinprincipletoapartofthesurfaceincomparisontotheforthcomingglobalvariantofthetheoremwhichappliestothewholesurfaceandinvolvestheEulercharacteristicandthegenusofthesurfacewhichareglobalfeaturesofthesurface.Anyway,theselabelsarenotofcrucialimportanceaslongasthetheoremanditssignificanceareunderstoodandappreciated.SomeexamplesfortheapplicationoftheaboveformoftheGauss-Bonnettheoremaregivenbelow:1.AdiscinaplanewithradiusRwhereEq.386↑becomes:(387)(1 ⁄ R)2πR + 0 + 0 = 2π + 0 + 0 ≡ 2πwhichisanidentity.2.Asemi-circulardiscinaplanewithradiusRwhereEq.386↑becomes:(388)[(1 ⁄ R)πR + 0 × 2R] + 2(π ⁄ 2) + 0 = π + π + 0 ≡ 2πwhichisanidentityagain.3.Asphericaltriangle(Fig.39↓)onasphereofradiusRwhosesidesaretwohalfmeridiansconnectingapoletotheequatorandonequarterofanequatorialparallelandallofitsthreecornersarerightangleswhereEq.386↑becomes:(389)[0(3{πR ⁄ 2})] + 3(π ⁄ 2) + (1 ⁄ R2)(4πR2 ⁄ 8) = 0 + (3π ⁄ 2) + (π ⁄ 2) ≡ 2π4.Theupperhalfofasphere(orahemisphereingeneral)ofradiusRwhereEq.386↑becomes:(390)0(2πR) + 0 + (1 ⁄ R2)2πR2 = 0 + 0 + 2π ≡ 2π

Figure39 Asphericaltrianglewiththreerightanglesonthesurfaceofasphere.Thethreesidesofthissphericaltrianglearearcsofgreatcircles.ThefactthatthesumoftheinterioranglesofaplanartriangleisequaltoπcanalsoberegardedasaninstanceoftheGauss-BonnettheoremsinceforaplanartriangleEq.386↑becomes:(391)0 + Σ3

i = 1(π − θi) + 0 = 3π − Σ3

i = 1θi = 2πwhereθiaretheinterioranglesofthetriangleandhenceΣ3

i = 1θi = πasitshouldbe.Byasimilarargument,wecanobtainthesumoftheinterioranglesofaplanarpolygonofnsides(n > 2)usingtheGauss-Bonnettheorem,thatis:(392)0 + Σn

i = 1(π − θi) + 0 = 

nπ − Σni = 1θi = 2π

whereθiaretheinterioranglesofthen-polygonandhenceΣni = 1θi = (n − 2)πasitshouldbe.

ThefactthattheperimeterofaplanarcircleofradiusRis2πRcanberegardedasanotherinstanceoftheGauss-BonnettheoremsinceforsuchaplanarcircleEq.386↑becomes:(393)(1 ⁄ R)L + 0 + 0 = 2πwhereListhelengthofthecircleperimeterandhenceL = 2πRwhichistherequiredresult.AsaresultoftheGauss-Bonnettheorem,thesumθsoftheinterioranglesofageodesictriangleonasurfacewithGaussiancurvatureofconstantsignis:1.θs < πiffK < 0.2.θs = πiffK = 0.3.θs > πiffK > 0.

Figure40 GeodesictrianglesonasurfacewithnegativeGaussiancurvature(leftframe),asurfacewithzeroGaussiancurvature(middleframe),andasurfacewithpositiveGaussiancurvature(rightframe).ThesecasesaredepictedinFig.40↑.ThisshowsthatthetotalcurvatureprovidestheexcessoverπforthesumwhenK > 0onthesurfaceandthedeficitwhenK < 0.ThevanishingtotalcurvatureinthecaseofK = 0istheintermediatecasewherethetotalcurvaturetermhasnocontributiontothesum.ThiscanbeseenfromEq.386↑which,forageodesictriangle,willreduceto:(394)0 + (3π − θs) + ∬DKdσ = 2π ⇒ θs = π + ∬DKdσWeremarkthat“geodesictriangle”isatrianglewithgeodesicsidesandhenceκg = 0identicallyoveritsboundary(see§5.7↓).Also,“triangle”hereandinthesphericaltriangleexamplerelatedtoFig.39↑ismoregeneralthanathree-sideplanarpolygonwiththreestraightsegmentsasitcanbeonacurvedsurfacewithcurvednon-planarsides.Asaconsequenceofthefindingsinthelastparagraph,twogeodesiccurvesonasimplyconnectedpatchofasurfacewithnegativeGaussiancurvaturecannotintersectattwopointsbecauseonintroducingavertexataregularpointononecurvewewillhaveanartificialcornerwithzeroexteriorangleandhenceπinteriorangle.Wewillthenhaveageodesictrianglewithθs > πonasurfaceoverwhichK < 0,inviolationoftheabove-statedcondition.Byasimilarargumenttotheargumentinthepreviousparagraph,theareaofageodesicpolygononasurfacewithconstantnon-zeroGaussiancurvatureisdeterminedbythesumofthepolygoninterioranglesθs.ThiscanalsobeseenfromEq.386↑whichinthiscasewillbereducedto:(395)0 + (nπ − θs) + K∬Ddσ = 2π ⇒ ∬Ddσ = [θs + (2 − n)π] ⁄ Kwheren > 2isthenumberofsidesofthepolygon.Asforgeodesictriangle,“geodesicpolygon”isapolygonwithgeodesicsidesandhenceκg = 0identicallyoveritsboundary.Again,“polygon”hereisgeneralandhenceitincludescurvilinearpolygononcurvedsurfacewithcurvednon-planarsides.Itisworthnotingthatbecausethegeodesiccurvatureisanintrinsicproperty,asdiscussedin§4.3↑,theGauss-Bonnettheorem,asgivenbyEq.386↑,isanotherindicationtothefactthattheGaussiancurvature(aswellasthetotalcurvature)isanintrinsicpropertyandhenceitisanotherdemonstrationofTheoremaEgregium(see§4.7↑).TheGauss-BonnettheoremhasalsoaglobalvariantwhichlinkstheEulercharacteristicχ,whichisatopologicalinvariantofthesurface,totheGaussiancurvatureK,whichisageometricinvariantofthesurface.ThisglobalformoftheGauss-Bonnettheoremstatesthat:onacompactorientablesurfaceSofclassC3thesetwoinvariantsarelinkedthroughthefollowingequation:(396)∬SKdσ = 2πχNow,sinceχisatopologicalinvariantofthesurface,Eq.396↑revealsthatthetotalcurvatureisalsoatopologicalinvariantofthesurface.TheglobalGauss-BonnettheoremcanbeusedtodeterminethetotalcurvatureKtofasurface.Forexample,theEulercharacteristicofasphereis2andhencefromEq.396↑itcanbeconcludedthatitstotalcurvatureisKt = 4πwithnoneedforevaluatingtheareaintegral.Similarly,theEulercharacteristicofatorusis0andhenceitcanbeconcludedimmediatelythatitstotalcurvatureisKt = 0withnoneedforevaluatingtheintegral.TheEulercharacteristicsofthesphereandtorusintheseexamplescanbeobtainedeasilybypolygonaldecomposition,asdescribedin§1.4.1↑.Forexample,theEulercharacteristicofthespherecanbecalculatedbydividingthesurfaceofthesphereto4curvedpolygonalfaceswith4verticesand6edgesandhencetheEulercharacteristicis:(397)χ = V + ℱ − ℰ = 4 + 4 − 6 = 2asseeninFig.16↑.TheglobalGauss-Bonnettheoremcanalsobeusedintheoppositedirection,thatisitmaybeusedfordeterminingtheEulercharacteristicofasurfaceknowingitsGaussian,andhencetotal,curvaturealthoughinmostcasesthismaybeoflittleusepractically.Forinstance,theGaussiancurvatureofasphereofradiusRis(1 ⁄ R2)ateverypointonthesphereandhenceitstotalcurvatureisKt = Kσ = (1 ⁄ R2)4πR2 = 4π,thereforefromEq.396↑itsEulercharacteristicisχ = (4π) ⁄ (2π) = 2.TheGauss-Bonnettheoremcanalsobeusedtofindthetotalcurvatureofasmoothsurfacewhichistopologically-equivalent(i.e.homeomorphic)toanothersurfacewithknowntotalcurvaturewithoutneedforanycalculation.Forexample,theellipsoid(Fig.4↑)ishomeomorphictothesphereandhencetheyhavethesameEulercharacteristic.Therefore,accordingtoEq.396↑theyhavethesametotalcurvaturewhichis4πasknownfromtheaforementionedsphereexample.Asaconsequence,thetotalcurvatureKtofasmoothsurfacewithacomplexshapecanbeobtainedfromtheGauss-

Bonnettheorembyreducingthesurfacetoatopologically-equivalentsimplersurfacewhosetotalcurvaturecanbeevaluatedpromptly.Asseenbefore(referto§1.4.1↑),foranorientablesurfaceofgenusgtheEulercharacteristicisgivenby:χ = 2(1 −  g),andhenceitstotalcurvatureisgivenby:(398)Kt = 2πχ = 4π(1 −  g)So,foracompactorientablecomplexly-shapedsurfacewhichcanbereducedtoaspherewith2handlesthetotalcurvatureisKt = 4π(1 − 2) =  − 4π.Similarly,thegenusofatorusisg = 1andhenceitstotalcurvatureisKt = 4π(1 − 1) = 0,asfoundearlierbyanothermethod.Hence,thetotalcurvatureofanycomplexly-shapedsurfacethatcanbereducedtoatorusiszero.TheimportantandobviousimplicationoftheglobalvariantoftheGauss-BonnettheoremthatcanbeconcludedfromthepreviousdiscussionisthatthetotalcurvatureofaclosedsurfaceisdependentonitsgenusandEulercharacteristicandnotonitsgeometricshapeandhenceitisatopologicalparameterofthesurfaceasstatedbefore.

4.9 LocalShapeofSurfaceUsingtheprincipalcurvatures,κ1andκ2,apointPonasurfaceisclassifiedaccordingtotheshapeofthesurfaceinthecloseproximityofPas:1.Flatwhenκ1 = κ2 = 0,andhenceK = H = 0(seeEqs.355↑and382↑).2.Parabolicwheneitherκ1 = 0andκ2 ≠ 0orκ2 = 0andκ1 ≠ 0,andhenceK = 0andH ≠ 0.3.Ellipticwheneitherκ1 > 0andκ2 > 0orκ1 < 0andκ2 < 0,andhenceK > 0.4.Hyperbolicwhenκ1 > 0andκ2 < 0,andhenceK < 0.Theseconstraintsonκ1andκ2,andhenceonKandH,aresufficientandnecessaryconditionsfordeterminingthetypeofthesurfacepointasdescribedabove.Thefollowingaresomeexamplesfortheaboveclassification:1.Thepointsofplaneareflat.2.Thepointsofcone(excludingtheapex)andthepointsofcylinderareparabolic.3.Thepointsofellipsoid(Fig.4↑)areelliptic.4.Thepointsofcatenoid(Fig.10↑)arehyperbolic.Surfacesnormallycontainpointsofdifferentshapes.Forexample,thetorushasellipticpointsonitsoutsidehalf,parabolicpointsonitstopandbottomparallels(seeFootnote23in§8↓),andhyperbolicpointsonitsinsidehalf,asseeninFig.38↑.However,therearesometypesofsurfacewhoseallpointsareofthesameshape;e.g.allpointsofplanesareflat,allpointsofspheresareelliptic,allpointsofcatenoidsarehyperbolic,andallpointsofcylindersareparabolic.Theaboveclassificationregardingthelocalshapecanalsobebasedonthedeterminantbofthecovariantcurvaturetensorandthecoefficientse, f, gofthesecondfundamentalformofthesurfacewhere:1.b = eg − f2 = 0ande = f = g = 0forflatpoints.2.b = eg − f2 = 0ande2 + f2 + g2 ≠ 0forparabolicpoints.3.b = eg − f2 > 0forellipticpoints.4.b = eg − f2 < 0forhyperbolicpoints.ConsideringEqs.356↑and383↑plusthefactthatthefirstfundamentalformispositivedefiniteandhencea > 0,thisclassificationwhichisbasedonbande, f, gisequivalenttothepreviousclassificationwhichisbasedonKandH.Theaboveclassificationoftheshapeofasurfaceintheimmediateneighborhoodofapoint(i.e.beingflat,parabolic,ellipticorhyperbolic)isaninvariantpropertywithrespecttopermissiblecoordinatetransformations.Thiscanbeconcludedfromthedependenceoftheclassificationonthesignofbasexplainedabove,plusEq.221↑wherethesquareoftheJacobian(whichisreal)ispositiveandhencethesignofbandbisthesame.Theclassificationisalsoindependentoftherepresentationandparameterizationofthesurfacesincethesepointtypesarerealgeometricpropertiesofthesurfaceintheirlocaldefinitions.Theinvarianceoftheshapetypeofthesurfacepoints,asexplainedinthepreviousstatements,holdstrueevenforthetransformationsthatreversethedirectionofthenormalvectortothesurface,n,becausetheclassificationdependsontheGaussiancurvaturewhichisinvariantevenunderthistypeoftransformations(referto§4.5↑).RegardingthedistinctionbetweentheflatandparabolicpointswhichinvolvesHaswell,thedistinctionisnotaffectedsinceitdependsonthemagnitudeofH(i.e.beingzeroornot)andnotonitssignandthemagnitudeisnotaffectedbysuchtransformations.IntheimmediateneighborhoodofanellipticpointPofasurfaceS,thesurfaceliescompletelyononesideofthetangentplanetoSatP(Fig.41↓),whileatahyperbolicpointthetangentplanecutsthroughSandhencesomepartsofSareononesideofthetangentplanewhileotherpartsareontheotherside(Fig.42↓).Intheneighborhoodofaparabolicpoint,thesurfaceliesentirelyononesideofthetangentplaneexceptforsomepointsonacurvewhichliesinthetangentplaneitself(Fig.43↓)(seeFootnote24in§8↓).Asforplanarpoints,theneighborhoodofthepointliesinthetangentplane.

Figure41 Tangentplaneatanellipticpoint.

Figure42 Tangentplaneatahyperbolicpoint.

Figure43 Tangentplaneataparabolicpoint.ThesurfacepointscanalsobeclassifiedaccordingtothegeometricshapeofDupinindicatrix(referto§3.6.1↑)asfollows:1.Ifeg−f2 = 0ande = f = g = 0,thenthepointisflatandtheDupinindicatrixisnotdefined.Hence,havingundefinedDupinindicatrixisacharacteristicforplanarpoints.Thenormalcurvatureatthepointiszeroinalldirections.2.Ifeg−f2 = 0ande2 + f2 + g2 > 0,theneitherκ1 = 0andκ2 ≠ 0orκ2 = 0andκ1 ≠ 0;hencethepointisparabolicandtheDupinindicatrixbecomestwoparallellines.Thepointischaracterizedbyhavingavanishingnormalcurvaturealongthedirectionoftheselineswhileithasthesamesigninallotherdirections.3.Ifeg−f2 > 0thenκ1andκ2havethesamesign;hencethepointisellipticandtheDupinindicatrixisanellipseorcircle.Thenormalcurvatureatthepointhasthesamesigninalldirections.4.Ifeg−f2 < 0thenκ1andκ2haveoppositesigns;hencethepointishyperbolicandtheDupinindicatrixbecomestwoconjugatehyperbolas.Thenormalcurvatureatthepointispositivealongthedirectionscorrespondingtooneofthesehyperbolasandnegativealongthedirectionscorrespondingtotheotherhyperbola,whilealongthecommonasymptotesofthesehyperbolasthenormalcurvatureiszero.Inbrief,becauseofthesecorrelationsbetweenthetypeofpointandtheshapeofitsDupinindicatrix,theDupinindicatrixcanbeusedtoclassifythepointasflat,parabolic,ellipticorhyperbolic.Itisworthnotingthattherelationbetweeneg−f2andκ1andκ2asstatedintheabovebulletpointscanbeconcludedfromthefactthat(seeEqs.355↑and356↑):(399)K = κ1κ2 = (eg−f2) ⁄ asinceaispositivedefinite.Weremarkthatintheimmediateneighborhoodofapointonasurface,thesurfacemaybeapproximatedby:1.Aplaneataflatpoint.2.Aparaboliccylinder(Fig.9↑)ataparabolicpoint.3.Anellipticparaboloid(Fig.7↑)atanellipticpoint.4.Ahyperbolicparaboloid(Fig.8↑)atahyperbolicpoint.AnotherremarkisthatintheneighborhoodofaparabolicpointPonasurfaceS,thetangentplaneofSatPmeetsSinasinglelinepassingthroughP,whileintheneighborhoodofahyperbolicpointPonasurfaceS,thetangentplanemeetsSintwolinesintersectingatPwherethesetwolinesdivideSalternativelyintoregionsabovethetangentplaneandregionsbelowthetangentplane.Finally,thefollowingfunction:(400)IIS ⁄ 2 = (edudu + 2fdudv + gdvdv) ⁄ 2evaluatedatagivenpointPofaclassC2surfacemaybecalledtheosculatingparaboloidofP.Thisosculatingparaboloid,representedbyhalfthesecondfundamentalform,isusedtodeterminetheshapeofthesurfaceatP(alsosee§4.2↑).

4.10 UmbilicalPointApointonasurfaceiscalled“umbilical”or“umbilic”or“navel”ifallthenormalsectionsofthesurfaceatthepointhavethesamenormalcurvatureκn.Hence,atumbilicalpointswehavethefollowingcondition:(401)κ1 = κ2Asstatedbefore,fornormalsectionsthenormalcurvatureκnisequalinmagnitudetothecurvatureκ.Therefore,thecurvatureofallthenormalsectionsatumbilicalpointsisalsoequal.TheconditionofEq.401↑impliesthatanumbilicalpointcannotbeahyperbolicpointbecauseathyperbolicpointsweshouldhaveκ1 > 0andκ2 < 0(referto§4.9↑).Hence,atumbilicalpointstheGaussiancurvatureshouldsatisfythenecessary(butnotsufficient)condition:K ≥ 0.Thefollowingaresomeexamplesofumbilicalpointsoncommonsurfaces:1.Allpointsofplanesareumbilical.However,someauthorsimposetheconditionK > 0atumbilicalpointsandhencethepointsofplanesarenotumbilicalaccordingtotheseauthors.2.Allpointsofspheresareumbilical.Hence,umbilicalpointsmaybecalledsphericalpoints.3.Thevertexofanellipticparaboloidofrevolutionisanumbilicalpoint.4.Thetwoverticesofanellipsoidofrevolutionareumbilicalpoints.IfallpointsofasurfaceofclassC3areumbilicalthenthesurfacemustbeasphere.Theplaneisaspecialcaseofsphereasitcanberegardedasaspherewithaninfiniteradius.AsufficientandnecessaryconditionforagivenpointPtobeumbilicalisthatthecoefficientsofthecurvaturetensorbαβatPareproportionaltothecorrespondingcoefficientsofthemetrictensoraαβatP,thatis:(402)bαβ = caαβwherec,whichisaproportionalityfactor,isindependentofthedirectionofthetangenttothenormalsectionattheumbilicalpointandα, β = 1, 2.Infact,thisconditionisthesameasthepreviouslystatedconditionofEq.321↑,andhencethesamejustificationofEq.321↑willapplyhere.Wealsonotethatc = κn,asseenthere.AsaresultofEq.402↑,atumbilicalpointsthedeterminantsofthetwotensors,aandb,satisfythefollowingrelation:(403)b = c2awherecissquaredbecauseaαβisatensorrepresentedbya2 × 2matrix.Now,sincethefirstfundamentalformispositivedefinite,andhencea > 0,thenifattheumbilicalpointc = 0thenb = 0accordingtoEq.403↑andthepointisaflatumbilic;otherwiseb > 0(sincecisreal)andthepointisanellipticumbilic(see§4.9↑)(seeFootnote25in§8↓).Onaplanesurfaceallpointsareflatumbilic,whileonasphereallpointsareellipticumbilic.Asseenearlier,ahyperbolicpointcannotbeanumbilicalpointsinceattheumbilicalpointweshouldhaveidenticalnormalcurvaturesinalldirections,bothinsignandinmagnitude,andthiscannothappenatahyperbolicpointwhoseκ1andκ2shouldbeofoppositesigns.Thisisinlinewiththefactthatatanumbilicalpointbcannotbenegativesincecisreal.Infact,Eq.403↑canberecastintothefollowingform:(404)K = b ⁄ a = c2 = (κn)2wheretheequation:c = κnandEq.356↑areused.Hence,alltheabove-statedfactsaboutthenatureoftheumbilicalpointandtheimpossibilityofbeinghyperbolic,asrepresentedbytheconditionK ≥ 0,arejustified.Becauseatumbilicalpointsκ1 = κ2,wehave:(405)K = κ1κ2 = κ1κ1 = (κ1)2 = [(2κ1) ⁄ 2]2 = [(κ1 + κ2) ⁄ 2]2 = H2

whereKandHaretheGaussianandmeancurvaturesatthepoint(see§4.5↑and4.6↑).ThiscanalsobeobtainedfromEq.348↑wherethediscriminantofthisquadraticequationbecomeszero(i.e.4H2 − 4K = 0),sinceatanumbilicalpointthetworootsareequal,andhenceH2 = K.ItshouldberemarkedthattheprovisionH2 = Kisasufficientandnecessaryconditionfora

pointatwhichthisconditionissatisfiedtobeumbilical.Thiscanbeconcludedfromthestatedrequirementsinthelastparagraph.AnotherremarkisthattherelationbetweenKandHatumbilicalpoints,asexpressedbyEq.405↑,maybestatedbysomeauthorsinthefollowingdisguisedform:(406)(aαβbαβ)2 = (4 ⁄ a)(b11b22 − b212)whereEqs.356↑and383↑areemployedinthisformandα, β = 1, 2.

4.11 ExercisesExercise1.Discussthesimilaritiesanddifferencesbetweenthecurvatureofcurvesandthecurvatureofsurfaces.Exercise2.Define,descriptivelyandmathematically,thecurvaturevectorKofsurfacecurvesanditsrelationtotheprincipalnormalvectorNofthecurve.Exercise3.ComparethevectorsnandNatapointonasurfacecurveoutliningtheirsimilaritiesanddifferences.Exercise4.Discussthedependencyofthecurvaturevectorofasurfacecurveatagivenpointofthecurveonthefollowingparameters:curveorientation,curveparameterization,surfaceorientationasindicatedbythedirectionofn,surfaceparameterization,tangentialdirectionandpositionofthepointonthesurface.Exercise5.What“inflectionpoint”onasurfacecurvemeans?Exercise6.Whatistheradiusofcurvatureatapointofinflection?Exercise7.Resolvethecurvaturevectorofasurfacecurveintoitstangentialandnormalcomponentsandnamethesecomponents.Expressthesecomponentsintermsoftheunitvectorsnanduexplainingallthesymbolsinvolvedinthisexpression.Exercise8.Findthecurvaturevector,K,ofaspacecurverepresentedby:r(t) = (3t2, t, 2sint).Exercise9.Define,descriptivelyandquantitatively,thenormalandgeodesiccurvaturesκnandκg.Exercise10.Whichofκnandκgisanintrinsicpropertyandwhichisanextrinsicproperty?Explainwhy.Exercise11.Comparethefollowingfourmovingframes:(T, N, B),(E1,  E2,  n),(n, T, u)and(d1,  d2,  n)outliningtheirsimilaritiesanddissimilarities.Exercise12.Whichoftheframesinthepreviousquestionemploybothsurfaceandcurvevectors?Whichoftheseframesareorthonormalbydefinitionandwhicharenot?Exercise13.Howthecurvatureκofasurfacecurveatagivenpointisrelatedtoitsnormalandgeodesiccurvaturesκnandκgatthatpoint?CanyoumakesenseofthisconsideringthenormalandtangentialcomponentsofthecurvaturevectorK?Exercise14.ProvethatthegeodesiccurvatureofanaturallyparameterizedcurveisgivenbyEq.325↑.Exercise15.ShowthatinanytwoorthogonaldirectionsatagivenpointPonasufficientlysmoothsurface,thesumofthenormalcurvaturescorrespondingtothesedirectionsatPisconstant.Exercise16.GiveabriefstatementofthetheoremofMeusnieroutliningitssignificance.Statethistheoreminasecondalternativeform.Exercise17.Showthattheosculatingcirclesofallcurvesonasurfacethatpassthroughagivenpointandinaspecificdirectionareonasphere.Exercise18.Define,descriptivelyandmathematically,thenormalcomponentKnofthecurvaturevectorKofasurfacecurveoutliningitsrelationtothecurvaturevectorandthenormalvectortothesurface,n.Exercise19.Define,descriptivelyandmathematically,thegeodesiccomponentKgofthecurvaturevectorKofasurfacecurveoutliningitsrelationtothecurvaturevectorandthesurfacebasisvectorsE1andE2.Exercise20.Derivetheformulaforthenormalcurvatureκnasaratioofthesecondfundamentalformtothefirstfundamentalform.Exercise21.Whythesignofthenormalcurvatureκnisdeterminedonlybythesignofthesecondfundamentalform?Exercise22.ShowthatatanypointPofasmoothsurfaceSthereexistsaparaboloidtangenttoSatPsuchthatthenormalcurvatureoftheparaboloidinanydirectionisequaltothenormalcurvatureofSatPinthatdirection.Exercise23.Discuss,indetail,thefollowingstatement:“Thenormalcurvatureatagivenpointonasurfaceandinagiventangentialdirectiontothesurfaceisapropertyofthesurface”.CanyoulinkthistotheMeusniertheorem?Exercise24.Forwhattypeofsurfacecurvethefollowingrelationistrue:|κn| = κ?Explainwhythisisso.Exercise25.Whatisthesignificanceofhavingaparaboloidatthepointsofasmoothsurfacewhosenormalcurvatureinagivendirectionisequaltothenormalcurvatureofthesurfaceinthatdirection?Exercise26.Whatisthesignofb(i.e.beinggreaterthan,lessthanorequaltozero)atflat,elliptic,parabolicandhyperbolicpointsonasurface,wherebisthedeterminantofthesurfacecovariantcurvaturetensor?Exercise27.ClassifythelocalshapeofasurfaceatagivenpointPaccordingtothevaluesofKandHatP.Exercise28.Usingoneofthemathematicaldefinitionsofthegeodesiccurvatureκg,explainwhyκgshouldbeclassifiedasanintrinsicorextrinsicproperty.Exercise29.Atwhattypeofsurfacepointsthefollowingrelationistrue:e ⁄ E = f ⁄ F = g ⁄ G = cwherecisconstantforalldirections?Whatcstandsfor?Exercise30.Expresstheequalitiesinthepreviousquestionintermsofthecoefficientsofthecovariantmetricandcovariantcurvaturetensors,aαβandbαβ,ofthesurface.Exercise31.OutlinetwodirectconsequencesofMeusniertheorem.Exercise32.WriteamathematicalrelationlinkingthegeodesiccomponentKgofthecurvaturevectortothesurfacebasisvectorsE1andE2.Exercise33.WhatistherelationbetweenKgandthetangentspaceTPSofthesurfaceatagivenpoint?Exercise34.Givetheformulaeofthegeodesiccurvatureκgofthecoordinatecurves.Simplifytheseformulaeinthecaseofhavingorthogonalcoordinatecurves.Exercise35.Stateamathematicalrelationbetweenthecurvatureκandthegeodesiccurvatureκgofasurfacecurveatagivenpointonthecurveexplainingallthesymbolsinvolved.Exercise36.Giveaformulaforthegeodesiccurvatureκginwhichextrinsicentitiesareinvolved.Doesthismeanthatκgisanextrinsicpropertyofthesurface?Exercise37.Explain,indetail,allthesymbolsusedinthefollowingformula:κg = (dθ ⁄ ds) + κgucosθ + κgvsinθWhatisthenameofthisformula?Exercise38.Provetherelationgiveninthelastquestion.Exercise39.GiveamathematicalformulainwhichκnisexpressedintermsofthecoefficientsofthefirstandsecondfundamentalformsE, F, G, e, f, g.Exercise40.Whatthetwo“principalcurvatures”ofasurfaceatagivenpointmean?Exercise41.Findanalyticalexpressionsfortheprincipalcurvaturesonasurfacerepresentedbytheequation:ξ2cosξ3 − ξ1sinξ3 = 0whereξ1, ξ2, ξ3arerealvariables.Exercise42.Theprincipalcurvaturesofasurfaceatagivenpointcorrespondtothetwodirectionsrepresentedbyλ1andλ2whicharetherootsofthefollowingquadraticequation:(gF − fG)λ2 + (gE − eG)λ + (fE − eF) = 0Fromtherulesofpolynomialequations,findthesumandproductoftheseroots.Exercise43.Definethe“principaldirections”descriptivelyandmathematically.Exercise44.Showthatκisaprincipalcurvaturewithaprincipaldirection(dv ⁄ du)iffthefollowingconditionsaresatisfied:(e − κE)du + (f − κF)dv = 0(f − κF)du + (g − κG)dv = 0Exercise45.Findtheprincipalcurvaturesandtheprincipaldirectionsonasurfacerepresentedparametricallyby:r(u, v) = (u, v, 2u2 + 5v2)atthepointwith(u, v) = (2.3, 1.6).Exercise46.ProveEulertheorem(seeEq.341↑andthesurroundingtext).Exercise47.WhatisDarbouxframe?Arethevectorsofthisframeorthonormal?Isthisframedefinedatumbilicalpointsonthesurface?Fullyjustifyyouranswerrelatedtothelasttwopartsofthequestion.Exercise48.Writetheformulaeforthepositionsofthecentersofcurvatureofthenormalsectionscorrespondingtothetwoprincipal

curvaturesatagivenpointonasurface.Exercise49.Correlate,mathematicallywithfullexplanationofallthesymbolsinvolved,thenormalcurvatureκnatagivenpointandinagivendirectiononasmoothsurfacetothetwoprincipalcurvaturesatthatpoint.Exercise50.Define,mathematicallyintermsoftheprincipalcurvatures,thefollowingterms:principalradii,meancurvatureandGaussiancurvature.Exercise51.Distinguishbetweenthe“totalcurvature”ofacurveandthe“totalcurvature”ofasurface.Forsurface,whatarethetwomeaningsofthisterm?Exercise52.FindtheGaussianandmeancurvaturesofasurfacegivenby:r(u, v) = (3u − v, u + 2v, 1.5uv)atthepointwith(u, v) = (3, 1).Exercise53.Statethelimitingconditionsontheprincipalcurvatures,andhencededucetheconditionsonthemeanandGaussiancurvatures,onthesurfaceofsphereandonthesurfaceofhyperboloidofonesheet.Exercise54.ProvethatthereisnocompactsurfaceofclassC2withnon-positiveGaussiancurvatureoverthewholesurface.Exercise55.Analyzethefollowingequationoutliningitssignificance:S(x, y)≃S(0, 0) + [(κ1x2) ⁄ 2] + [(κ2y2) ⁄ 2]Exercise56.Statethenecessaryandsufficientconditionforarealnumbertobeaprincipalcurvatureofasurfaceatagivenpoint.Exercise57.Investigatethenumberofrootsofthefollowingquadraticequationandtheimpactofthisonthenumberofprincipalcurvaturesofthesurfaceatthepointwherethisequationapplies:(EG − F2)κ2 − (gE − 2fF + eG)κ + (eg − f2) = 0Exercise58.Fromtheequationinthepreviousquestion,obtainananalyticalexpressionfortheprincipalcurvaturesofthesurfaceatthepointwherethisequationapplies.Exercise59.Fromtheequationinthelasttwoquestions,obtaintheequation:κ2 − 2Hκ + K = 0andhenceverifythattheprincipalcurvaturesaregivenby:κ1, 2 = H±√(H2 − K).Exercise60.Writedowntheequationsoftheprincipalcurvatureswhentheu1andu2coordinatecurvesarealignedalongtheprincipaldirections.Exercise61.ObtainEq.367↑fortheGaussiancurvatureofasurfacewithorthogonalcoordinatecurvesbyusingEq.366↑.Exercise62.Showthatthespheresaretheonlyconnected,compactandsufficientlysmoothsurfaceswithconstantGaussiancurvature.Exercise63.StatethecurvatureformulaofRodriguesdefiningallthesymbolsinvolvedanddiscussingitssignificance.Exercise64.TestthevalidityoftheRodriguesformulafortheprincipaldirectionsatthepointwith(u, v) = (1.4, 3.9)onasurfaceparameterizedby:r(u, v) = (u, v, u2 + 3v2).Exercise65.UsetheRodriguescurvatureformulatoprovethatspheresaretheonlyconnectedclosedsurfacesofclassC3whoseallpointsaresphericalumbilical.Exercise66.GiveamathematicalexpressionfortheGaussiancurvatureintermsofthecoefficientsofthesurfacemetricandcurvaturetensors.Exercise67.Whatisthesignificanceofhavinganintrinsicsurfacecurvature,representedusuallybytheGaussiancurvature,asawayfora2Dinhabitanttohavesomeperceptionofthenatureofthesurfaceanditsshapeasseenfromtheambientspacebya3Dinhabitant?Exercise68.Discussthefollowingstatement:“TheGaussiancurvaturealonganyparallellineofasurfaceofrevolutionisconstant”.Exercise69.Startingfromthefollowingrelation:K = (b ⁄ a),derivetherelation:K = det(bαβ).Exercise70.GiveamathematicalrelationcorrelatingtheGaussiancurvaturetothefollowingcoefficientsofthe2DRiemann-Christoffelcurvaturetensor:R1212,R1221,R1121andR2112.Exercise71.WhatistheGaussiancurvatureofaMongepatchoftheformr(u, v) = (u, v, f(u, v))?Exercise72.WhytheGaussiancurvatureisindependentoftheorientationofthesurface(whereorientationisbasedonthechoiceofthedirectionoftheunitnormalvectortothesurface)?Exercise73.WhichofthefollowinggeometricshapeshaveidenticalGaussiancurvaturesattheircorrespondingpointsandwhy:plane,sphere,cylinder,catenoid,ellipsoid,hyperbolicparaboloid,helicoid,andcone?Compare,inyouranswer,eachpairoftheseshapes.Exercise74.WritedownanexpressionfortheGaussiancurvatureofasurfaceofrevolutiongeneratedbyrevolvingasufficientlydifferentiableplanecurveoftheformx = f(y)aroundthey-axis.Exercise75.Explainallthesymbolsofthefollowingequationwithdiscussionofitssignificanceinrelationtotheintrinsicandextrinsicgeometriesofthesurface:∂un × ∂vn = K(E1 ×  E2)Exercise76.StatethemathematicalexpressionthatcorrelatestheGaussiancurvaturetotheRiccicurvaturescalarofasurface.Exercise77.UsingEq.367↑andtheparametricequationsofBeltramipseudo-sphere(Eqs.43↑-45↑),showthatthepseudo-spherehasanegativeconstantGaussiancurvatureandfindthiscurvature.Exercise78.ClassifysurfaceswithregardtotheirGaussiancurvatureashavingconstantorvariablecurvaturegivingtwoexamplesforeach.Exercise79.WhatistheimpactofscalingasurfaceupordownbyaconstantpositivefactoronitsGaussiancurvature?Exercise80.DiscusstheeffectofanisometricmappingofasurfaceonitsGaussiancurvature.Exercise81.StatetheHilbertlemmagivingexamplesforitsapplicationsfromcommontypesofsurface.Exercise82.Givetheconditionsforthevalidityofthefollowingequation:

Also,giveitssimplifiedforminthecaseofrepresentingthesurfacebygeodesiccoordinatesstatingtheotherconditionsrequiredforthissimplification.Exercise83.ExpressthemeancurvatureasafunctionoftheGaussiancurvaturetakingcareofthesigns.Exercise84.ShowthatspheresaretheonlyconnectedcompactsurfaceswithconstantmeancurvatureandpositiveGaussiancurvature.

Exercise85.ExplainhowthepositionofthesurfaceinadeletedneighborhoodofagivenpointPrelativetothetangentplaneofthesurfaceatPisusedtoclassifythenatureoftheGaussiancurvatureatP.Fromthisperspective,discussthesignoftheGaussiancurvatureonthepointsofthefollowingsurfaces:hyperbolicparaboloid,sphere,torusandcylinder.Exercise86.TheGaussiancurvatureofadevelopablesurfaceisidenticallyzero.Why?Exercise87.WhatistheGaussiancurvatureofasurfaceparameterizedby:x = (5 + cosφ)cosθ,y = (5 + cosφ)sinθandz = sinφ?Exercise88.Provideamathematicaldefinitionforthetotalcurvatureofasurfaceexplainingallthesymbolsusedinthedefinition.Exercise89.Defineallthesymbolsusedinthefollowingequation:εαβεγδRαβγδ = KεαβεγδεαβεγδExercise90.ExplainindetailhowthefollowingequationimpliesthattheGaussiancurvatureisarank-0tensor:K = (1 ⁄ 4)εαβεγδRαβγδ.Exercise91.WritetheGaussiancurvatureintermsofthesurfacecurvaturetensorusingthemostsimpleform.Exercise92.AlgebraicallymanipulatetherelationK = (b ⁄ a)toobtainthefollowingrelation:K = [(∂uE1⋅ E1 ×  E2)(∂vE2⋅ E1 ×  E2) − (∂vE1⋅ E1 ×  E2)2] ⁄ (EG − F2)2Exercise93.ExpressthemeancurvatureHintermsofthecoefficientsofthefirstandsecondfundamentalforms.Exercise94.WhatistherelationbetweenthemeancurvatureHandthemixedtypesurfacecurvaturetensorbβα?Exercise95.ComparethesignofthemeancurvaturetothesignoftheGaussiancurvaturewithregardtotheirdependencyonthedirectionoftheunitnormalvectortothesurface.Exercise96.Givetwoexamplesofcommontypesofsurfaceoverwhichthemeancurvatureisconstant.Also,giveanexampleofasurfacewithvariablemeancurvature.Exercise97.WhatisthemeancurvatureofaMongepatchoftheformr(u, v) = (u, v, f(u, v))?Exercise98.WhatistheessenceofGaussTheoremaEgregium?Giveanexampleofanequationoratheoremthatdemonstratesthistheorem.Exercise99.DeriveEq.385↑usingEq.380↑.Exercise100.WritedownthemathematicalequationrepresentingthelocalformoftheGauss-Bonnettheoremexplainingallthesymbolsinvolved.Exercise101.GiveanexamplefortheapplicationofthelocalGauss-Bonnettheoremusingaplanargeometricshapeandanotherexampleusinganon-planarshape.Exercise102.ExplainwhytwogeodesiccurvesonapatchofasurfacewithnegativeGaussiancurvaturecannotintersectattwopoints.Exercise103.ApplytheGauss-BonnettheoremonthesphericaltriangleofFig.39↑givingdetailedexplanationsforeachstep.Exercise104.UseacircularflatdisctodemonstratetheapplicationofthelocalformoftheGauss-Bonnettheoremgivingdetailedexplanationsforeachstep.Exercise105.WhatistheglobalformoftheGauss-Bonnettheoremandwhatisitssignificancegeometricallyandtopologically?Exercise106.FindthetotalcurvatureofthesurfacesdepictedinFig.17↑.Exercise107.Show,mathematically,thattheareaofageodesicpolygononasurfacewithconstantnon-vanishingGaussiancurvatureisdeterminedbythesumoftheinternalanglesofthepolygon.Exercise108.Verifythatthetotalcurvaturesofellipsoidandtorusarerespectively4πand0byperformingdetailedsurfaceintegralcalculations.Exercise109.OutlinetheusefulnessoftheglobalformoftheGauss-Bonnettheoreminobtainingthetotalcurvatureofasurfacewithknowntopologicalpropertieswithoutperformingdetailedcalculations.Exercise110.UsingtheGauss-Bonnettheorem,provethattheGaussiancurvatureisidenticallyzeroonasurfaceSifatanypointPonStherearetwofamiliesofgeodesiccurvesintheneighborhoodofPintersectingataconstantangle.Exercise111.WritedownthemathematicalrelationthatlinkstheEulercharacteristicofasurfacetoitstopologicalgenus.Exercise112.UsetheprincipalcurvaturesandthemeanandGaussiancurvaturestoclassifythepointswithregardtothelocalshapeofthesurfaceasflat,elliptic,parabolicandhyperbolicgivingexamplesofcommongeometricshapesforeachcase.Exercise113.Repeattheclassificationofthepreviousquestionusingthistimethecoefficientsofthesecondfundamentalformofthesurface.Exercise114.Provethatonacircularcylinderallpointsareparabolic.Exercise115.Showthatintheneighborhoodofanellipticpointonasurface,thesurfaceliesononesideofitstangentplaneatthatpoint.Exercise116.Asurfaceisrepresentedparametricallyby:r(u, v) = (u, v, u2 + v3).Determinetheconditionsthatidentifytheparabolic,hyperbolicandellipticpointsonthesurface.Exercise117.Giveanexampleofasurfacehavingelliptic,parabolicandhyperbolicpointsatdifferentlocations.Exercise118.Whythepointtype(i.e.beingflat,elliptic,hyperbolicorparabolic)onasurfaceisaninvariantpropertywithrespecttochangesinthesurfacerepresentationandparameterization?Exercise119.Whythepointtypeisinvariantwithrespecttoachangeofthesurfaceorientationbyreversingthedirectionofthenormalvectortothesurface?Exercise120.Makeasimplesketchoutliningthepositionofasurfacerelativetothetangentplaneatelliptic,parabolicandhyperbolictangencypoints.Exercise121.Demonstratethatthesurfacerepresentedparametricallyby:r(u, v) = (u, v, u2 + v3)liesonbothsidesofitstangentplaneatthepoint(u, v) = (0, 0).Exercise122.Forasurfacerepresentedparametricallyby:r = (u, v, v4),findtheequationofacurveonthesurfacewhosepointshaveacommontangentplane.Exercise123.DescribehowDupinindicatrixcanbeusedtoclassifythepointsofasurfacewithregardtothelocalshape(i.e.flat,elliptic,parabolicandhyperbolic).Exercise124.Whataretheprototypicalgeometricshapesthatprovidethebestapproximationforthelocalshapeofasufficientlysmoothsurfaceatits:flat,elliptic,hyperbolicandparabolicpoints?Exercise125.What“umbilicalpoint”means?Whataretheothertermsusedtolabelsuchapoint?Exercise126.Whatarethecharacteristicfeaturesofumbilicalpoints?Exercise127.Givefiveexamplesofumbilicalpointsoncommongeometricsurfacessuchasspheresandparaboloids.Exercise128.Statethemathematicalrelationbetweenthecoefficientsofthemetricandcurvaturetensorsatumbilicalpoints.Exercise129.Demonstratethatatanumbilicalpointofasurfacewehave:K = H2whereKandHaretheGaussianandmeancurvaturesatthepoint.Exercise130.Showthattherelation:K = H2canalsobewrittenas:(aαβbαβ)2 = (4 ⁄ a)(b11b22 − b212)Exercise131.Explainwhyatumbilicalpointswehaveb = c2awhereaandbarethedeterminantsofthecovariantmetricandcovariantcurvaturetensorsandcisaproportionalityfactor.Exercise132.Givetwoexamplesofsurfaceswhoseallpointsareumbilical,andtwootherexamplesofsurfaceswithnoumbilicalpointatall.Also,giveanexampleofasurfacewithonlyoneumbilicalpoint,andanotherexampleofasurfacewithonlytwoumbilicalpoints.

Chapter5SpecialCurvesTherearemanyclassificationstospacecurvesdependingontheirpropertiesandtheirrelationswitheachother.Inthefollowingsectionsofthischapter,webrieflyinvestigateafewofthesecategories.

5.1 StraightLineAnecessaryandsufficientconditionforacurveofclassC2tobeastraightlineisthatitscurvatureiszeroateverypointonthecurve.Hence,anothercriterionforacurvetobeastraightlineisthatallthetangentsofthecurveareparallel,where“parallel”hereisusedinitsabsoluteEuclideansense(see§2.7↑).AnothercriterionforacurveC(t):I → ℝ3wheret ∈ I ⊆ ℝtobeastraightlineisthatforallpointstinthedomainofthecurve,ṙandrarelinearlydependentwherer(t)isthespatialrepresentationofthecurveandtheoverdotsrepresentderivativewithrespecttothegeneralparametertofthecurve.Astraightlinelyingonasurfacehasthesametangentplaneateachofitspoints,andhencethelineiscontainedinthisuniquetangentplane.Anystraightlineonanysurfaceisageodesiccurve(see§5.7↓)andanasymptoticline(see§5.9↓).

5.2 PlaneCurveAcurveisdescribedasaplanecurveifthewholecurvecanbecontainedinaplanewithnodistortion(Fig.44↓).Anecessaryandsufficientconditionforacurveparameterizedbyageneralparameterttobeaplanecurveisthattherelationṙ⋅(r × r) = 0holdsidenticallywherer(t)isthespatialrepresentationofthecurveandtheoverdotsrepresentdifferentiationwithrespecttot.Planecurvesarecharacterizedbyhavingidenticallyvanishingtorsion.Infact,havingidenticallyvanishingtorsionisanecessaryandsufficientconditionforaregularcurveofclassC2tobeaplanecurve.Forplanecurves,theosculatingplaneateachregularpointonthecurvecontainstheentirecurve.Therefore,theplanecurvemaybecharacterizedbyhavingacommonintersectionpointforallofitsosculatingplanes.Italsoimpliesthatthecurvehasthesameosculatingplaneatallofitspoints.Twocurvesareplanecurvesiftheyhavethesamebinormallinesateachpairoftheircorrespondingpoints.ThelocusofthecentersofcurvatureofacurveCisanevolute(see§5.3↓)ofCiffCisaplanecurve.Onasmoothsurface,ageodesiccurve(see§5.7↓)whichisalsoalineofcurvature(see§5.8↓)isaplanecurve.AplanecurvehasalwaysaBertrandcurveassociate(see§5.4↓).

Figure44 Planecurve.

5.3 InvoluteandEvoluteIfCeisaspacecurvewithatangentsurfaceST(see§6.6↓)andCiisacurveembeddedinSTanditisorthogonaltoallthetangentlinesofCeattheirintersectionpoints,thenCiiscalledaninvoluteofCewhileCeiscalledanevoluteofCi(seeFig.45↓).Hence,theinvoluteisanorthogonaltrajectoryofthegeneratorsofthetangentsurfaceofitsevolute.Accordingly,theequationofaninvoluteCitoacurveCeisgivenby:(407)ri = re + (c − s)Tewhereriisanarbitrarypointontheinvolute,reisthepointonthecurveCecorrespondingtori,cisagivenconstant,sisanaturalparameterofCeandTeistheunitvectortangenttoCeatre.

Figure45 EvoluteCe,involuteCi,tangentlines(dashed)andtangentsurface(shaded).AvisualdemonstrationofhowtogenerateaninvoluteCiofacurveCe,when(c − s)inEq.407↑ispositive,maybegivenbydetachingatautstringattachedtoCewherethestringiskeptinthetangentdirectionasitisdetached.AfixedpointPonthestring,wherethedistancebetweenPandthepointofcontactofthestringwithCerepresentsanaturalparameterofCe,thentracesaninvoluteofCe.AcurvehasinfinitelymanyinvolutescorrespondingtodifferentvaluesofcinEq.407↑.Therefore,theinvolutesmaybedescribedasparallelcurvesonthetangentsurface.Similarly,aninvolutehasaninfinitenumberofevolutescorrespondingtodifferentvaluesofc.Foranytangentofagivencurve,thelengthofthelinesegmentconfinedbetweentwogiveninvolutesisconstantwhichisthedifferencebetweenthetwoc’sinEq.407↑ofthetwoinvolutes.IfCeisanevoluteofCi,thenforagivenpointPeonCeandthecorrespondingpointPionCitheprincipalnormallineofCeatPeisparalleltothetangentlineofCiatPi.AcurveCiisaplanecurveiffthelocusofthecentersofcurvatureofCiisanevoluteofCi.Theinvolutesofacirclearecongruent.Theevolutesofplanecurvesarehelices.

5.4 BertrandCurveBertrandcurvesaretwoassociatedspacecurveswithcommonprincipalnormallinesattheircorrespondingpoints.AssociatedBertrandcurvesarecharacterizedbythefollowingproperties:1.Theproductofthetorsionsoftheircorrespondingpointsisconstant,thatis:(408)τ1(so)τ2(so) = constantwhereτ1andτ2arethetorsionsofthetwocurvesandsoisagivenvalueoftheircommonparameter.2.Thedistancebetweentheircorrespondingpointsisconstant.3.Theanglebetweentheircorrespondingtangentlinesisconstant.ForaplanecurveC1,thereisalwaysacurveC2suchthatC1andC2areassociatedBertrandcurves.IfC1isacurvewithnon-vanishingtorsionsuchthatC1hasmorethanoneBertrandcurveassociate,thenC1isacircularhelix.Thereverseisalsotrue.IfC1isacurvewithnon-vanishingtorsionthenanecessaryandsufficientconditionforC1tobeaBertrandcurve(i.e.itpossessesanassociatecurveC2suchthatC1andC2areBertrandcurves)isthattherearetwoconstantsc1andc2suchthat:(409)κ = c1τ + c2whereκandτarethecurvatureandtorsionofthecurveC1.IfC1andC2aretwoinvolutesofaplanecurveC,thenC1andC2areBertrandcurves.

5.5 SphericalIndicatrixAsphericalindicatrixofacontinuously-varyingunitvectorisacontinuouscurveContheorigin-basedunitspheregeneratedbymappingtheunitvector(e.g.TorNorB)ofaparticularspacecurveConanequalunitvectorrepresentedbyapointontheorigin-basedunitsphere.Hence,wehaveCT,CNandCBasthesphericalindicatricesofCcorrespondingrespectivelytothetangent,principalnormalandbinormalvectorsofC(seeFootnote26in§8↓).Figure46↓isasimpledemonstrationofthesphericalindicatrixCTofaspacecurveC.

Figure46 ThesphericaltangentindicatrixCTofaspacecurveCwherethenumbersindicatethecorrespondencebetweentheunittangentvectorsofCandtheirmaponCT.IfC(s)isanaturallyparameterizedcurvethenswillnotnecessarilybeanaturalparameterforthetangentindicatrixCT.AnecessaryandsufficientconditionforstobeanaturalparameterforCTisthatκ(s) = 1identicallywhereκisthecurvatureofC.ThetangenttothecurveCTofacurveCisparalleltothenormalvectorNofCatthecorrespondingpointsofthetwocurves.ThetangenttothecurveCTofacurveCisalsoparalleltothetangenttothecurveCBofCatthecorrespondingpointsofthetwocurves.ThenecessaryandsufficientconditionforthecurveCTofacurveCtobeacircleisthatCisahelix.ThecurvatureofthecurveCTofacurveCisrelatedtothecurvatureandtorsionofCby:(410)(κT)2 = (κ2 + τ2) ⁄ κ2

whereκTisthecurvatureofCTwhileκandτarethecurvatureandtorsionofCrespectively.ThetorsionofthecurveCTofanaturallyparameterizedcurveCisgivenby:(411)τT = (κ’τ − κτ’) ⁄ [κ(κ2 + τ2)]whereτTisthetorsionofCT,κandτarethecurvatureandtorsionofCrespectively,andtheprimestandsforderivativewithrespecttothenaturalparametersofC.ThecurvatureofthecurveCBofacurveCisgivenby:

(412)κB = (κ2 + τ2) ⁄ κ2

whereκBisthecurvatureofCBwhiletheothersymbolsareasexplainedbefore.ThetorsionofthecurveCBofanaturallyparameterizedcurveCisgivenby:(413)τB = (κ’τ − κτ’) ⁄ [τ(κ2 + τ2)]whereτBisthetorsionofCB.

5.6 SphericalCurveAsphericalcurveisacurvethatliescompletelyonthesurfaceofasphere.Sphericalindicatricesarecommonexamplesofsphericalcurves(see§5.5↑).Circlesaretheonlysphericalcurveswithconstantcurvature.Atallpointsofasphericalcurve,thenormalplaneofthecurvepassesthroughthecenteroftheembeddingsphere.Conversely,ifallthenormalplanesofacurvemeetinacommonpoint,thenthecurveissphericalwiththecommonpointbeingthecenterofthespherethatenvelopsthecurve.Thesufficientandnecessaryconditionthatshouldbesatisfiedbyasphericalcurveisgivenby:(414)

whereRκandRτaretheradiiofcurvatureandtorsionandsisanaturalparameterofthecurve.ThecenterofcurvatureofatwistedsphericalcurveCatagivenpointPonCistheprojectionofthecenteroftheenvelopingsphereontheosculatingplaneofCatP.

5.7 GeodesicCurveThecharacteristicfeatureofageodesiccurveisthatithasvanishinggeodesiccurvatureκgateverypointonthecurve.Thisisanecessaryandsufficientconditionforasurfacecurvetobegeodesic.Inmoretechnicalterms,letS:Ω → ℝ3beasurfacedefinedonasetΩ ⊆ ℝ2andletC(t):I → ℝ3,whereI ⊆ ℝ,bearegularcurveonS,thenCisageodesiccurveiffκg(t) = 0onallpointst ∈ Iinitsdomain.ThepathoftheshortestdistanceconnectingtwopointsinaRiemannianspaceisageodesic.Thelengthofarc,asgivenbyEq.204↑,isusedinthedefinitionofgeodesicinthissense.Aphysicalinterpretationmaybegiventothegeodesiccurvethatafreeparticlerestrictedtomoveonthesurfacewillfollowageodesicpath.Anotherphysicalinterpretationisthatageodesicpathminimizesthetotalkineticenergyspentbyamassiveobjectinmovingbetweentwopointswhenthepathistraversedwithconstantspeed.Thesetwophysicalinterpretationsmayrestonthesamephysicalprinciple.ThegeodesicisastraightlineinaEuclideanspace,butitisageneralizedcurvedpathinageneralRiemannianspace.IfageodesicsurfacecurveisnotastraightlinethenitsprincipalnormalvectorNiscollinearwiththenormalvectorntothesurfaceateachpointonthecurvewithnon-vanishingcurvature;theoppositeisalsotrue.Infact,acurveonasurfaceisgeodesiciffitiseitherastraightlineoritsprincipalnormalvectoriscollinearwithnoverthewholecurve.Asstatedbefore,collinearityofNandnisequivalenttotheconditionthatnliesintheosculatingplaneofthecurveatthegivenpoint.Anothersufficientandnecessaryconditionforacurvetobeageodesiccurveisthatthefirstvariation(see§1.4.2↑)ofitslengthiszero.Infact,thismaybetakenasthebasisforthedefinitionofgeodesicasthecurveconnectingtwofixedpoints,P1andP2,whoselengthpossessesastationaryvaluewithregardtosmallvariationsinitsneighborhood,thatis:(415)

ItcanbeshownthatageodesiccurvesatisfiestheEuler-Lagrangevariationalprinciple(see§1.4.2↑)whichisanecessaryandsufficientconditionforextremizingthearclength.Figure47↓isanillustrationofhowthelengthofthegeodesiccurvebetweentwogivenpointsissubjecttothevariationalprinciple.

Figure47 Thelengthofageodesiccurve(solid)connectingtwopoints,P1andP2,asanextremumwithrespecttothelengthofothercurves(dashed)connectingthesepointsthatresultfromsmallperturbationsinitsneighborhood.Examplesofgeodesiccurvesonsimplesurfacesarethearcsofgreatcirclesonspheres.Infact,beinganarcofagreatcircleisasufficientandnecessaryconditionforbeingageodesiccurveonasphere.Otherexamplesofgeodesiccurvesarethearcsofhelices,thegeneratingstraightlinesandthecirclesoncylinders(Fig.48↓).Thegeneratingstraightlinesandthecirclesoncylindersmaybeconsideredasdegeneratehelices.Themeridiansofasurfaceofrevolutionarealsogeodesics.Thearcsofparallelcirclesonasurfaceofrevolutioncorrespondingtostationarypointsonthegeneratingcurveofthesurfacearealsogeodesiccurves.Allstraightlinesonanysurfacearegeodesiccurves.Forplanesurfacesinparticular,beingastraightlineonaplaneisasufficientandnecessaryconditionforbeingageodesic.Thelinesofcurvature(see§5.8↓)arealsogeodesiccurves.

Figure48 Thethreetypesofgeodesiccurvesoncylinders(a)circulararcs(b)generatinglinesand(c)helicalarcs.Intrinsically,thegeodesiccurvesarestraightlinesinthesensethata2Dinhabitantwillseethemstraightsincehecannotdetecttheircurvature.Thisisduetothefactthatonlythegeodesicpartofthecurvatureisanintrinsicpropertyandhenceitcanbedetectedbya2Dinhabitant,thereforeifthispart

ofthecurvaturevanishedthe2Dinhabitantwillfailtodetectanycurvaturetothecurvewhichisequivalentforhimtohavingastraightline.Anydeviationfromsuch“straightlines”withinthesurfaceisthereforeageodesiccurvatureandhenceitcanbedetectedintrinsicallybya2Dinhabitant.Althoughageodesiccurveisfrequentlythecurveoftheshortestdistancebetweentwopointsonthesurfaceitisnotnecessarilyso.Forinstance,thelargestofthetwoarcsformingagreatcircleonasphereisageodesiccurvebutitisnotthecurveoftheshortestdistanceonthespherebetweenitstwoendpoints;infactitisthecurveofthelongestdistanceamongthecirculararcsconnectingthetwopoints(Fig.49↓).Asimilarexampleisthetwoarcsofaparallelcircleonacircularcylinderconnectingtwopointswherethetwoarcsaredifferentinlength.Anyway,ifonasurfaceSthereisexactlyonegeodesiccurveconnectingtwogivenpoints,P1andP2,thenthelengthofthegeodesiccurvesegmentbetweenP1andP2istheshortestdistanceonSbetweenthesepoints.

Figure49 Twogeodesiccurvesconnectingtwopoints,P1andP2,onthesurfaceofasphere:ashortonebetweenP1andP2directly,andalongonebetweenP1andP2throughP3.Bothofthesegeodesicsarearcsofagreatcircleonthesphere.Basedonthepreviousstatements,beingashortestpathisasufficientbutnotnecessaryconditionforbeingageodesic,thatisallshortestpathsconnectingtwogivenpointsaregeodesicsbutnotallgeodesicsareshortestpaths.Aconstraintmaybeimposedtomakethecriterionofminimallengthapplytoallgeodesicsbystatingthatgeodesicsminimizedistancelocallybutnotnecessarilygloballywhereaninfinitesimalelementofarcisconsideredinthisconstraint.Anyway,theuniversalcriterionthatshouldbeadoptedtoidentifygeodesiccurvesisthevanishingofthegeodesiccurvatureoverthewholecurve,asstatedatthestartofthissection.

Thegeodesic,eveninitsrestrictedsenseasthecurveoftheshortestdistance,isnotnecessarilyunique;forexampleallsemi-circularmeridiansoflongitudeconnectingthetwopoles(orinfactallsemi-circulararcsconnectinganytwoantipodalpoints)ofaspherearegeodesicseveninthatsenseandthereisaninfinitenumberofthem.Infact,eventheexistence,notonlyuniqueness,ofageodesicconnectingtwopointsonasurfaceisnotguaranteed.Anexampleisthexyplaneexcludingtheoriginofcoordinateswithtwopointsonastraightlinelyingintheplaneandpassingthroughtheoriginwherethereisalowerlimitforthelengthofanycurveconnectingthetwopoints(Fig.50↓).Thislimitisthestraightlinesegmentconnectingthetwopointsbutthissegmentcannotbeageodesicontheplanesinceitincludestheoriginwhichisnotontheplane.AnycurveC(otherthanthestraightlinesegment)ontheplaneconnectingthetwopointscannotbeacurveofshortestlength,andhenceageodesic,sincethereisalwaysanothercurveontheplaneconnectingthetwopointswhichisshorterthanC.Inthiscontext,wenotethatonaplanesurfaceallgeodesiccurvesarestraightlinesandhenceofshortestlength.

Figure50 Non-existenceofageodesiccurveconnectingthetwopointsP1andP2ontheshownxyplanewhichdoesnotincludetheoriginofcoordinatesO.ThedashedlinerepresentsthestraightlinesegmentconnectingP1andP2whilethesolidlineCrepresentsothercurvesontheplanebetweenthetwopoints.IntheneighborhoodofagivenpointPonasurfaceandforanyspecificdirection,thereisexactlyonegeodesiccurvepassingthroughPinthatdirection.Moretechnically,foranyspecificpointPonasurfaceSofclassC3,andforanytangentvectorvinthetangentspaceofSatP,thereexistsageodesiccurveonthesurfaceinthedirectionofvthatpassesthroughP.Infact,thisisbasedontheexistenceofauniquesolutiontothegeodesicdifferentialequations(Eqs.416↓-417↓)wheninitialvaluesofapointonthecurveanditsderivative(whichrepresentsitstangentdirection)atthatpointaregiven.Anobviousexampleofthepreviousstatementistheplanewhereastraightlinepassesthroughanypointandinanydirection.Anotherexampleisthespherewhereagreatcirclepassesthroughanypointandinanydirection.Alessobviousexampleisthecylinderwhereahelix(includingthestraightlinegeneratorsandthecircleswhichcanberegardedasdegenerateformsofhelix)passesthroughanypointandinanydirection.Similarly,thereisexactlyonegeodesiccurvepassingthroughtwosufficientlyclosepointsonasmoothsurface(seeFootnote27in§8↓).Asindicatedbefore,geodesicsincurvedspacesaretheequivalentofstraightlinesinflatspaces.Forplanes(orinfactforanyEuclideannDmanifold)thereexistsauniquegeodesicpassingbetweenanytwopoints(whetherthetwopointsarecloseornot)whichisthestraightlinesegmentconnectingthetwopoints.

Thenecessaryandsufficientconditionthatshouldbesatisfiedbyanaturallyparameterizedcurveonasurface,bothofclassC2,tobeageodesiccurveisgivenbythefollowingsetofsecondordernon-lineardifferentialequations:(416)(d2u1 ⁄ ds2) + Γ111(du1 ⁄ ds)2 + 2Γ112(du1 ⁄ ds)(du2 ⁄ ds) + Γ122(du2 ⁄ ds)2 = 0(417)(d2u2 ⁄ ds2) + Γ211(du1 ⁄ ds)2 + 2Γ212(du1 ⁄ ds)(du2 ⁄ ds) + Γ222(du2 ⁄ ds)2 = 0wheresisthearclength,andtheChristoffelsymbolsarederivedfromthesurfacemetric.Thelastequationscanbemergedinasingleequationusingtensornotation,thatis:(418)δ(duα ⁄ ds) ⁄ δs ≡ (d2uα ⁄ ds2) + Γαβγ(duβ ⁄ ds)(duγ ⁄ ds) = 0whereα, β, γ = 1, 2andthestandardnotationofabsolutederivativeisinuse(see§7↓).Theseequations,whichcanbeobtainedfromEq.323↑bysettingthetwocomponentsofthegeodesiccurvaturevectortozero,havenoclosedformexplicitsolutionsingeneralbecauseoftheirnon-linearity.SimilarequationsareusedtoidentifythegeodesiccurvesingeneralnDspaces.FromEq.418↑,itcanbeseenthatbeingageodesicisanintrinsicpropertysincetheconditionsrepresentedbythisequationdependexclusivelyontheChristoffelsymbolswhichdependonlyonthecoefficientsofthefirstfundamentalformandtheirpartialderivatives.Hence,geodesiccurvescanbedetectedandmeasuredbya2Dinhabitant.FromEq.418↑,itcanalsobeseenthatforplanes(orindeedforanyEuclideannDmanifold)thegeodesicisastraightlinesinceinthiscasetheChristoffelsymbolsvanishidenticallyandEq.418↑willbereducedto(d2uα ⁄ ds2) = 0whichhasastraightlinesolution.FromEq.326↑,weseethattheu1coordinatecurvesonasufficientlysmoothsurfacearegeodesicsiffΓ211 = 0.Similarly,fromEq.328↑,weseethattheu2coordinatecurvesaregeodesicsiffΓ122 = 0.WealsoseefromEqs.327↑and329↑thatforcoordinatesystemswithorthogonalcoordinatecurves,thecoordinatecurvesaregeodesicsiffEisindependentofvandGisindependentofu.ForaMongepatchoftheformr(u, v) = (u, v, f(u, v)),thegeodesicdifferentialequationsaregivenby:(419)(1 + fu2 + fv2)u’’ + fufuu(u’)2 + 2fufuvu’v’ + fufvv(v’)2 = 0(420)(1 + fu2 + fv2)v’’ + fvfuu(u’)2 + 2fvfuvu’v’ + fvfvv(v’)2 = 0wherethesubscriptsuandvrepresentpartialderivativesoffwithrespecttothesurfacecoordinatesuandv,andtheprimerepresentsderivativeswithrespecttoanaturalparameter.Basedonwhatwehaveseensofar,itcanbeconcludedthateachoneofthefollowingprovisionsisanecessaryandsufficientconditionforacurveConasurfaceStobeageodesiccurve:1.Thegeodesiccomponentofthecurvaturevectoriszeroateachpointonthecurve,thatisKg = 0identically.Thisisbasedonthedefinitionofgeodesiccurvewhichwestatedearlier.2.TheosculatingplaneofthecurveateachpointofthecurveisorthogonaltothetangentplaneofSatthatpoint.Thereasonisthatongeodesiccurvesκg = 0andhencenandNareinthesameorientation(paralleloranti-parallel)onallpointsalongthecurve(refertoEqs.305↑and307↑)andhencenliesintheosculatingplaneandtheosculatingplanewillbeorthogonaltothetangentplane.3.Thenormalvectorntothesurfaceatanypointonthecurveliesintheosculatingplane.Thisisbecauseforageodesiccurve,KgvanishesidenticallyandhenceK = κN = κnn = Kn.4.TheprincipalnormalvectorNofCisnormaltothesurfaceateachpointonCsinceNiscollinearwithn.5.ThecurvaturevectorKofthecurveisnormaltothetangentplaneofthesurfaceateachpointonthecurve.Beingageodesicisindependentofthechoiceofthecoordinatesystemandhenceitisinvariantunderpermissibletransformations.Itisalsoindependentofthetypeofrepresentationandparameterizationandhenceitisinvariantinthissense.Geodesiccurvescanbeopenorclosedcurvesandmaybeself-intersecting.Inthisstatement,weareconsideringthetotalityofthegeodesicpathascharacterizedbyhavingidenticallyvanishinggeodesiccurvatureandnotasaconnectingarcbetweentwodistinctpoints(seeFootnote28in§8↓).Examplesofopengeodesicsarethestraightlinesonplanesandthehelicesoncylinderswhileexamplesofclosedgeodesicsarethegeodesicsofsphereswhicharegreatcirclesandtheparallelcirclesoncylinders.Infact,allthegeodesicsonsphereareclosedcurvesastheyaregreatcircles,whilecirclesoncylinderistheonlycaseofclosedgeodesicsonthistypeofsurface.AsaresultoftheGauss-Bonnettheorem(see§4.8↑),onasurfacewithnegativeGaussiancurvaturetwogeodesicscannotintersectatmorethanonepointifthegeodesicsencloseasimply-connectedregion.ThereasonisthatonintroducinganartificialvertexataregularpointononeofthesecurveswewillhaveanewcornerwithπinteriorangleandhencethesumoftheanglesofthegeodesictrianglewillexceedπwhichisimpossibleonasurfacewithK < 0.Also,onintroducinganartificialvertexataregularpointoneachoneofthesecurveswewillhaveageodesicquadrilateralwhoseinternalanglesadduptomorethan2πonasurfacewithK < 0whichisnotpossible(see§4.8↑).AnotherresultoftheGauss-BonnettheoremisthatasurfacewithnegativeGaussiancurvaturecannothaveageodesicthatintersectsitself.Thismaybeestablishedbyasimilarargumenttothepreviousonethatis:onintroducingtwoartificialcornersattworegularpointsonthecurve,wewillhaveageodesictrianglewhoseinterioranglesadduptomorethanπwhichisnotpossibleonasurfacewithK < 0(see§4.8↑).OnapatchofasurfaceofclassC2withorthogonalcoordinatecurvesandwiththefirstfundamentalformcoefficientsbeingdependentononlytheucoordinatevariable(i.e.E = E(u),F = 0andG = G(u))thefollowingstatementsapply:1.Theucoordinatecurvesaregeodesics.2.Thevcoordinatecurvesaregeodesicsiff∂uG = 0alongthesecurves.3.AcurveCrepresentedbyr = r(u, v(u))isageodesiciff:(421)

wherekisaconstant.Thecaseofdependenceononlythevcoordinatevariablecanbeobtainedbyre-labelingthecoordinatevariablesandcoefficients.Thesecondoftheabovestatementsmaybegeneralizedbysaying:onasurfacewithorthogonalcoordinatecurves,thecurvesofconstantuαaregeodesicsiffaββ(β ≠ α)isafunctionofuβonly.Asindicatedbefore,geodesicsincurvedspacesrepresentageneralizationofstraightlinesinflatspaces.Hence,geodesicsmaybedescribedasthestraightestcurvesinthespace.Infact,geodesiccurvesonadevelopablesurfacebecomestraightlineswhenthesurfaceisdevelopedintoaplanebyunrolling.Thismaybedemonstratedbytheperceptionofa2Dinhabitantofthesurfacewhowillfailtoobserveanydifferencetothegeodesiccurvewhenthesurfaceisdevelopedintoaplaneandthegeodesiccurvenecessarilybecomesastraightlineontheplane.Moregenerally,ageodesiccurvewillbemappedontoageodesiccurvebyanyisometrictransformationduetotheinvarianceofthegeodesiccurvatureunderthistypeoftransformationssincegeodesiccurvatureisanintrinsicproperty.Therefore,isometricsurfacespossessidenticalgeodesicequations.Anothersufficientandnecessaryconditionforasurfacecurvetobegeodesicisbeingatangenttoaparallelvectorfield.Avectorattainedbyparallelpropagation(see§2.7↑)ofatangentvectortoageodesiccurvestaysalwaystangenttothegeodesiccurve.Asaresult,avectorfieldattainedbyparallelpropagationalongageodesicmakesaconstantanglewiththegeodesic.

5.8 LineofCurvatureA“lineofcurvature”isacurveConasurfaceSdefinedonanintervalI ⊆ ℝasC:I → SwiththeconditionthatthetangentofCateachpointonCiscollinearwithoneoftheprincipaldirections(see§4.4↑)ofthesurfaceatthatpoint.Wenotethat“line”heredoesnotmeanstraight.Sincethedefinitionofthelineofcurvatureisseeminglybasedontheexistenceofdistinctprincipaldirections,umbilicalpoints(see§4.10↑)maybeexcludedfromtheabovedefinitionofthelineofcurvatureduetotheabsenceofdistinctprincipaldirectionsatthesepointsalthoughthereseemstobenoharminincludingisolatedumbilicalpoints(atleast)overthepathofthelineofcurvature(seeFootnote29in§8↓).ReferringtoEq.134↑,onalineofcurvatureeithersinθ = 0orcosθ = 0andhencethelinesofcurvaturearecharacterizedbyhavingidenticallyvanishinggeodesictorsion(i.e.τg = 0).Theconditionthatshouldbesatisfiedbyalineofcurvatureisusuallygivenbythefollowingrelation:(422)(a12b11 − a11b12)du1du1 + (a22b11 − a11b22)du1du2 + (a22b12 − a12b22)du2du2 = 0wheretheindexedaandbarethecoefficientsofthesurfacecovariantmetricandcovariantcurvaturetensorsrespectively.Infact,thisisthesameastheconditiongivenbyEq.350↑fortheprincipaldirections,whichisconsistentwiththefactthatthelineofcurvatureisalignedalongaprincipaldirectionateachofitspoints.Theconditionthatshouldbesatisfiedbyalineofcurvatureonasurfacemaybegivenintensornotationby:(423)εγδaαγbβδduαduβ = 0whereεγδisthe2Dabsolutepermutationtensor.ExamplesoflinesofcurvaturearemeridiansandparallelsofsurfaceofrevolutionofclassC2.Foradevelopablesurface,thelinesofcurvatureconsistofitsgeneratorsandtheirorthogonaltrajectories.Onasufficientlysmoothsurface,anygeodesicwhichisaplanecurveisalineofcurvature.Similarly,onasufficientlysmoothsurface,ifageodesiccurveCisalineofcurvaturethenCisaplanecurve.Thelinesofintersectionofeachpairofatriplyorthogonalsystemarealsolinesofcurvature.WeremarkthatthreefamiliesofsurfacesinasubsetVofa3DspaceformatriplyorthogonalsystemifateachpointPofVthereisasinglesurfaceofeachfamilypassingthroughPsuchthateachpairofthesesurfacesintersectorthogonallyattheircurveofintersection.Atanon-umbilicalpointPonasufficientlysmoothsurfaceS,theu1andu2coordinatecurvesarealignedwiththeprincipaldirectionsifff = F = 0atP.The“if”partcanbeseen,forexample,fromEq.350↑whichinthiscase(i.e.f = F = 0)willreduceto(gE − eG)du1du2 = 0andhenceitwillbesatisfiedonthecoordinatecurves,sincedu2willvanishontheu1coordinatecurvewhiledu1willvanishontheu2coordinatecurve,andthesecurvesbecomealignedwiththeprincipaldirections.The“onlyif”partcanalsobeseenfromEq.350↑becauseifthecoordinatecurvesarealignedwiththeprincipaldirectionsthenthisequationshouldbesatisfiedwhereitbecomes(fE − eF)du1du1 = 0fortheu1coordinatecurveand(gF − fG)du2du2 = 0fortheu2coordinatecurveandbothoftheseequationsimplyf = F = 0sincedu1 ≠ 0ontheu1coordinatecurveanddu2 ≠ 0ontheu2coordinatecurve.WenotethatbyconsideringthestatedconditionsandthedefinitionsofthecoefficientsofthefirstandsecondfundamentalformsaswellastheRodriguescurvatureformula(see§4.4↑),itcanbeconcludedthatthecoefficientsE, eandg, Gcannotvanish.Asaconsequenceofthelastparagraph,thecoordinatecurvesonthesurfaceS,excludingtheumbilicalpoints,arelinesofcurvatureifff = F = 0overtheentiresurface.Thismayalsobestatedbysayingthatonasmoothsurface,excludingplanesandspheres(whoseallpointsareumbilical),ifthelinesofcurvatureareselectedasthenetofcoordinatecurvesthena12 = b12 = 0overtheentiresurfaceexcludingtheumbilicalpoints.Weremarkthatwhentheuandvcoordinatecurvesofasurfacepatcharelinesofcurvature,theprincipalcurvatures,κ1andκ2,overtheentirepatchwillbegivenby:(424)κ1 = e ⁄ Eκ2 = g ⁄ GwhereE, G, e, garethecoefficientsofthefirstandsecondfundamentalformsatthepointsofthepatch.Thereaderisreferredto§4.4↑forjustification(seeEq.354↑andrelatedtext).OnasurfaceofclassC3,therearetwoperpendicularfamiliesoflinesofcurvatureintheneighborhoodofanynon-umbilicalpoint.Ifthecurveofintersectionoftwosurfacesisalineofcurvatureforonesurfacethenitisalineofcurvaturefortheothersurfacewhenthetwo

surfacesareintersectingeachotherataconstantangle.Thelinesofcurvatureformarealorthogonalgridoverthesurface.Acurveisalineofcurvatureiffthetangenttothecurveandthetangenttoitssphericalimage(see§3.10↑)attheircorrespondingpointsareparallel.Thelinesofcurvatureonasurface,whichisnotasphereorminimalsurface(see§6.7↓),arerepresentedbyanorthogonalnetonitssphericalimage.Asindicatedabove,intheneighborhoodofanon-umbilicalpointonasufficientlysmoothsurfacetherearetwoorthogonalfamiliesoflinesofcurvature.Hence,ateachpointPonsuchasurfaceacoordinatepatchincludingPcanbeintroducedintheneighborhoodofPwherethecoordinatecurvesatParealignedwiththeprincipaldirections.OnasurfacepatchwheretheGaussiancurvaturedoesnotvanish,theanglesbetweentheasymptoticlines(see§5.9↓)arebisectedbythelinesofcurvature.

5.9 AsymptoticLineAnasymptoticdirectionofasurfaceatagivenpointPisadirectionforwhichthenormalcurvaturevanishes,i.e.κn = 0.Hence,inanasymptoticdirectionatapointonasurfacewehave(seeEq.307↑):(425)K = Kg = κguAsaconsequenceofEq.316↑,κniszerointhedirectionsforwhichthesecondfundamentalformiszero.Hence,thenecessaryandsufficientconditionfortheasymptoticdirectionsisthat:(426)bαβduαduβ = b11(du1)2 + 2b12du1du2 + b22(du2)2 = 0WenotethatasymptoticdirectionsaredefinedonlyatpointsforwhichtheGaussiancurvatureisnon-positive(K ≤ 0)andhenceitisnotdefinedatellipticpoints(see§4.9↑)whereK > 0.Thisisbecauseatellipticpointseitherκ1 > 0andκ2 > 0orκ1 < 0andκ2 < 0andhenceκncannottakethevaluezeroatthesepoints.Asaresult,thenumberofasymptoticdirectionsatelliptic,parabolicandhyperbolicpointsis0,1and2respectively,whileatflatpointsalldirectionsareasymptotic.Thetwoasymptoticdirectionsofahyperbolicpointseparatethedirectionsofpositivenormalcurvaturefromthedirectionsofnegativenormalcurvature.Thesignofthenormalcurvatureatellipticandparabolicpointsisthesameinalldirections,excludingtheasymptoticdirectionoftheparabolicpoint.Similarly,atflatpointsthenormalcurvatureiszeroinalldirections.At-parameterizedsurfacecurveC(t):I → S,whereI ⊆ ℝisanopenintervalandSrepresentsthesurface,isdescribedasanasymptoticlineifateachpointt ∈ IthevectorT,whichisthetangenttothecurve,iscollinearwithanasymptoticdirectionatthatpoint.Itshouldberemarkedthat“line”hereisnotrequiredtobestraight;henceasymptoticlinesarealsocalledasymptoticcurves.Fromtheabovestatements,itcanbeseenthattheasymptoticlinesarecharacterizedbythefollowingfeatures:1.Thenormalcomponentofthecurvaturevectoriszeroateachpointonthecurve,thatisKn = 0identically.Thisisbasedonthedefinitionofasymptoticlineasstatedabove.2.Thetangentplanetothesurfaceateachpointofthecurvecoincideswiththeosculatingplaneofthecurveatthatpoint.Thisisaconsequenceofhavingidenticallyvanishingnormalcurvature,sincethecurvaturevectorwillthenhaveonlyatangentialcomponentandhencetheosculatingplaneateachpointofanasymptoticlinebecomestangenttothesurfaceatthatpoint.Thedifferentialequationrepresentingasymptoticlinescanbeobtainedfromtheconditionthatthenormalcurvaturevanishesidenticallyovertheline,thatis:(427)e(du1 ⁄ ds)2 + 2f(du1 ⁄ ds)(du2 ⁄ ds) + g(du2 ⁄ ds)2 = 0whichisbasedonEqs.316↑and233↑oronEq.318↑.Thenecessaryandsufficientconditionfortheu1andu2coordinatecurvestobecomeasymptoticlinesisthate = 0identicallyontheu1coordinatecurveandg = 0identicallyontheu2coordinatecurve(seeFootnote30in§8↓).ThisisbasedonEqs.319↑and320↑whicharefullyjustifiedthere.AccordingtoEq.308↑,κn = n⋅KwherenandKarerespectivelythenormalvectortothesurfaceandthecurvecurvaturevector.Hence,acurveonasufficientlysmoothsurfaceisanasymptoticlineiffn⋅K = 0identically.Now,sincethevectorncannotvanishontheregularpointsofthesurface,thenthisconditionisrealizedifateachpointonthecurveeitherK = 0orKandnareorthogonalvectors.Intheformercasethepointisaninflectionpointwhileinthelattercasetheosculatingplaneistangenttothesurfaceatthepoint.Therefore,allpointsonanasymptoticlineshouldbeoneofthesetypesortheother.Thereverseisalsotrue,i.e.acurvewhoseallpointsareoneofthesetypesortheotherisanasymptoticline.Asaresultofthelaststatements,anystraightlineonasurfaceisanasymptoticlinesincethecurvecurvaturevectorKvanishesidenticallyonsuchaline.AccordingtothetheoremofBeltrami-Enneper,alonganasymptoticnon-straightlineonasufficientlysmoothsurfacethesquareofthetorsionτisequaltothenegativeoftheGaussiancurvatureK,thatis:(428)τ2 =  − KwhereτandKareevaluatedateachindividualpointalongthecurve.SinceasymptoticdirectionsaredefinedonlyatpointsforwhichK ≤ 0,thesquareofthetorsionintheaboveequationisequaltotheabsolutevalueoftheGaussiancurvatureatthepoint,thatis:τ2 = |K|andhenceτisrealasitshouldbe.Thetorsionsoftwoasymptoticlinespassingthroughagivenpointonasufficientlysmoothsurfaceareequalinmagnitudeandoppositeinsign.

Aswewillsee(referto§5.10↓),asymptoticdirectionsareself-conjugate.Infact,someauthorstakeself-conjugationasthedefiningcharacteristicforbeingasymptotic.FromthedefinitionoftheasymptoticdirectionplustheEulerequation(Eq.341↑),weseethattheangleθwhichanasymptoticdirectionmakeswiththeprincipaldirectionofκ1atagivennon-umbilicalpointPonasufficientlysmoothsurfaceSisgivenby:(429)tan2θ =  − κ1 ⁄ κ2whereκ1andκ2aretheprincipalcurvaturesofSatP.Whenκ2 = 0,thereciprocalofthisrelationshouldbetaken.Again,tanθisrealsinceatpointswithK < 0,thetwokappasshouldhaveoppositesigns(seeEq.355↑).Thesituationissimilarwhenκ1 = 0andκ2 < 0.However,whenκ1 > 0andκ2 = 0thereciprocalwillbetakenandthecotangentwillbereal.Thepossibilityofκ1 = κ2 = 0isalreadyexcludedbythenon-umbilicalconditionsinceapointwithκ1 = κ2 = 0isaflatumbilic.BecauseEq.426↑isquadratic,itpossessestwosolutionswhicharerealanddistinct,orrealandcoincident,orconjugateimaginarydependingonitsdiscriminantΔwhichisoppositeinsigntothedeterminantbofthesurfacecovariantcurvaturetensor(seeFootnote31in§8↓).Hence,theasymptoticdirectionsatagivenpointonasurfacecanbeclassifiedaccordingtothedeterminantbatthepointas:1.RealanddistinctforΔ > 0andhenceb < 0.2.RealandcoincidentforΔ = 0andhenceb = 0.3.ConjugateimaginaryforΔ < 0andhenceb > 0.Thisisinlinewiththeabovestatementthatthenumberofasymptoticdirectionsatelliptic,parabolicandhyperbolicpointsis0,1and2respectivelybecause,asseenin§4.9↑,thelocalshapeofasurfaceatagivenpointisdeterminedbythesignofbatthepointwhereb < 0athyperbolicpoint,b = 0atparabolicandflatpoints,andb > 0atellipticpoint.WealsoremarkthatfromEq.356↑wecanseethatthesignoftheGaussiancurvatureKisthesameasthesignofbduetothefactthata > 0sincethefirstfundamentalformispositivedefinite,asestablishedearlier.Hence,theabove-describedclassificationoftheasymptoticdirectionscanalsobebasedonK,asstatedforbinthepreviouspoints.Again,asseenin§4.9↑,thesignofKisusedtodeterminethelocalshapeatapoint,andhencethenumberoftheasymptoticdirectionsisdeterminedaccordingly.ItisworthnotingthatonasufficientlysmoothsurfacewithorthogonalfamiliesofasymptoticlinesthemeancurvatureHiszero.ThiscanbeseenfromEq.383↑wherebyaligningthecoordinatecurvesalongtheasymptoticdirectionswithaproperlabeling,wewillget:F = 0sincethecoordinatecurvesareorthogonal,ande = g = 0accordingtotheaboveconditionwhichwestatedafterEq.427↑,andhenceH = 0.NowsinceHisinvariant,thenthiswillremainvalidunderpermissibletransformationstoothersurfacecoordinates.Anothernoteisthattheprincipaldirectionsatagivenpointonasmoothsurfacebisecttheasymptoticdirectionsatthepoint,asindicatedbefore.Also,onasmoothsurfaceofclassC3,therearetwodistinctfamiliesofasymptoticdirectionsintheneighborhoodofanyhyperbolicpoint.Asseenbefore,anystraightlinecontainedinasurfaceisanasymptoticline.Aswellasthepreviouslystatedexplanation,thismayalsobejustifiedbythefactthatsuchalineiswhollycontainedinaplane,whichisthetangentspaceofeachofitspoints,andhencethelineisanasymptoticlinewithanidenticallyvanishingnormalcurvature.

5.10 ConjugateDirectionAdirection(δv ⁄ δu)atapointonasufficientlysmoothsurfaceisdescribedasconjugatetothedirection(dv ⁄ du)iffthefollowingrelationholdstrue(seeFootnote32in§8↓):(430)dr⋅δn = 0where:(431)dr = E1du + E2dvδn = ∂unδu + ∂vnδvFromEq.244↑,itcanbeseenthattheconditiongivenbyEq.430↑isequivalenttothefollowingcondition:(432)eduδu + f(duδv + dvδu) + gdvδv = 0ThelastequationmayalsobeobtaineddirectlybysubstitutingfromEq.431↑intoEq.430↑andperformingthedotproductwiththeuseofEqs.249↑-251↑toobtainthecoefficientse, f, g.Duetothesymmetryintheaboverelations,(dv ⁄ du)isalsoconjugateto(δv ⁄ δu),andhencethetwodirectionsaredescribedasconjugatedirections.Atahyperbolicoranellipticpointonasufficientlysmoothsurface,eachdirectionhasauniqueconjugatedirection.Asstatedearlier,anasymptoticdirectionisaself-conjugatedirection.Twofamiliesofcurvesonasufficientlysmoothsurfacearedescribedasconjugatefamiliesifthedirectionsoftheirtangentsateachintersectionpointofthecurvesareconjugatedirections.Theuandvcoordinatecurvesonasmoothsurfaceareconjugatefamiliesofcurvesifff,whichisthecoefficientofthesecondfundamentalform,vanishesidentically.ThiscanbeseenfromEq.432↑wherebyaproperlabelingoftheuandvcoordinatesinthetwodirectionstomakeduδu = dvδv = 0onthecoordinatecurvesthefirstandlasttermsofEq.432↑willvanishonthecoordinatecurvesandhencethecurveswillbeconjugatefamiliesbysatisfyingthereducedconditionf(duδv + dvδu) = 0whichleadstotheconditionf = 0sinceduδv + dvδu ≠ 0accordingtothislabeling.Similarly,iff = 0thenthecoordinatecurveswillsatisfythereducedconditionf(duδv + dvδu) = 0andhencetheyareconjugatefamilies.

5.11 ExercisesExercise1.Statetwocriteriaforaspacecurvetobestraight.Exercise2.Provethatacurverepresentedbyr(t)isastraightlineifṙandrarelinearlydependentoverthewholecurve.Exercise3.Showthataspacecurvewhosealltangentlinesareparallelisastraightline.Exercise4.Correct,ifnecessary,thefollowingstatement:“Allstraightlinesonasurfacearegeodesiccurvesandviceversa”.Exercise5.Whatisthecharacteristicfeatureofplanecurves?Fromthis,explainwhythetorsionofplanecurvesisidenticallyzero.Exercise6.Provethatacurveisaplanecurveifitsosculatingplaneshaveacommonintersectionpoint.Exercise7.Showthataspacecurverepresentedbyr(t)isaplanecurveiffṙ⋅(r × r)vanishesidentically.Exercise8.Provethathavinganidenticallyvanishingtorsionisanecessaryandsufficientconditionforacurvetobeaplanecurve.Exercise9.Showthattwocurvesareplanecurvesiftheyhavethesamebinormallinesateachpairoftheircorrespondingpoints.Exercise10.Define,rigorously,involuteandevolutecurvesmakingasimpleplottooutlinetheirrelation.Alsoexplaintheroleofthetangentsurfaceoftheevoluteinthiscontext.Exercise11.Explainallthesymbolsusedinthefollowingequationwhichisrelatedtoinvolutecurves:ri = re + (c − s)Te.Makesenseofthisequationusingyourplotinthepreviousexercise.Exercise12.Outlinethevisualdemonstrationwhichiscommonlyusedtoexplaintherelationbetweenaninvoluteanditsevolute.Usetheplotmentionedinthelasttwoquestionsinyourexplanation.Exercise13.Showthatthetangentlineofacurveandtheprincipalnormallineofitsevoluteareparallelattheircorrespondingpoints.Exercise14.Provethattheevolutesofplanecurvesarehelices.Exercise15.ProvethatforaplanecurveCthelocusofthecentersofcurvatureofCisanevoluteofC.Exercise16.Derivetheparametricequationoftheinvoluteofacirclerepresentedby:r(θ) = (5cosθ, 5sinθ)where0 ≤ θ < 2π.Exercise17.ProvethatanytwoinvolutesofaplanecurveareassociatedBertrandcurves.Exercise18.Howmanyinvolutesagivencurvecanhave?Howtheseinvolutesarerelatedtoeachotherthroughtheconstantc(seeEq.407↑)?Exercise19.Howmanyevolutesagivencurvecanhave?Howtheseevolutesarerelatedtoeachotherthroughtheconstantc(seeEq.407↑)?Exercise20.Justifythefactthattheinvolutesofacirclearecongruentwithaclearexplanationofhowtheseinvolutesarerelatedtoeachother.Exercise21.DefineBertrandcurvesoutliningtwooftheirmaincharacteristicfeatures.Exercise22.ShowthatahelixhasaninfinitenumberofBertrandassociatesandidentifytheseassociates.Exercise23.ProvethatonapairofBertrandcurves,theanglebetweentheirtangentsatcorrespondingpointsisconstant.Exercise24.StateasufficientandnecessaryconditionforacurvetobeaBertrandcurvebyhavinganassociateBertrandcurve.Exercise25.ShowthataplanecurvehasalwaysaBertrandassociate.Exercise26.ProvethattheproductoftorsionsatthecorrespondingpointsofapairofassociatedBertrandcurvesisconstant(seeEq.408↑andrelatedtextforexplanation).Exercise27.ProvethatonapairofBertrandcurves,thedistancebetweentheircorrespondingpointsisconstant.Exercise28.GiveabriefdefinitionofsphericalindicatrixwithasimplesketchofthesphericalnormalindicatrixCNofaspacecurvetoillustratethisconcept.Exercise29.Provethefollowingequation(Eq.410↑):(κT)2 = (κ2 + τ2) ⁄ κ2.Exercise30.DiscussthesimilaritiesanddifferencesbetweenGaussmapping(see§3.10↑)andsphericalindicatrixmapping.

Exercise31.Justify,usingasimplefactabouthelices,thatthesphericalimagesofT, N, Bofahelixrotatingaroundthez-axisarecirclescenteredaroundthez-axis.Exercise32.Justifythefollowingstatement:“Thebinormalindicatrixisasinglepointforaplanecurve,andthetangentindicatrixisasinglepointforastraightline”.Exercise33.Prove,rigorously,thatthetangentindicatrixofahelixisacircle.Exercise34.ProvethatthetangenttothesphericalindicatrixofthetangenttoagivenspacecurveCandtheprincipalnormalofCareparallel.Exercise35.Writedownthemathematicalformulaforthetorsionofthesphericalbinormalindicatrixofaspacecurveexplainingallthesymbolsusedintheformula.Exercise36.What“sphericalcurve”means?giveacommonexampleofasphericalcurve.Exercise37.State,mathematically,thesufficientandnecessaryconditionforacurvetobeasphericalcurveexplainingallthesymbolsinvolved.Exercise38.Investigateifthecurverepresentedparametricallyby:r(t) = (5cost, 5costsint, 5sin2t)isasphericalcurveornot.Exercise39.ShowthatthesphericalimageofacurveC(t)isaclosedcurvewhenthevectorthatgeneratesthesphericalimageisaperiodicfunctionoftalthoughCmaynotbeperiodic.Discussinthiscontextthehelixasanexample.Exercise40.Whatisthecharacteristicfeatureofgeodesiccurves?Exercise41.Givearigorousmathematicaldefinitionofgeodesiccurve.Exercise42.Giveexamplesofgeodesiccurvesonthefollowingsurfaces:plane,sphereandcylinder.Exercise43.Onasurfaceofrevolution,whattypeofcurveisnecessarilygeodesicandwhattypeispotentiallygeodesic?Exercise44.Definegeodesiccurvevariationallyusingtheconceptsofcalculusofvariations.Exercise45.Showthatanyhelixonacircularcylinderisageodesiccurve.Exercise46.Outlinetherelationbetweentheconceptofgeodesiccurveandtheconceptofcurveofshortestdistancebetweentwopoints.Exercise47.ProvethatEq.418↑isasufficientandnecessaryconditionforacurvetobegeodesic.Exercise48.Findananalyticalexpressionrepresentingthegeodesiccurvesonacircularconeusingoneofitsparametricrepresentations.Exercise49.Outlinetheconceptofgeodesiccurveonasurfaceasperceivedbya2Dinhabitantofthesurface.Exercise50.Provethatallthegeodesiccurvesonaplanearestraightlines.Exercise51.Doesageodesiccurvenecessarilyexistbetweentwogivenpointsonaspacesurface,andifitdoesexististhegeodesiccurvenecessarilyunique?Supportyouranswerwithillustratingexamplesforbothcases.Exercise52.Discussthefollowingstatementanditsimplications:“Beingashortestpathisasufficientbutnotnecessaryconditionforbeingageodesiccurve”.Exercise53.Correct,ifnecessary,thefollowingstatement:“IntheneighborhoodofagivenpointPonasurface,thereisexactlyonegeodesiccurvethatpassesthroughP”.Exercise54.Writedown,withfullexplanation,thedifferentialequationswhichprovidethenecessaryandsufficientconditionsforanaturallyparameterizedcurveonasurfacetobegeodesic.Exercise55.CorrectthefollowingrelationswhichrepresentthegeodesicdifferentialequationsforaMongepatchoftheformr(u, v) = (u, v, f(u, v)):(1 + fu2 + fv2)u’ + fufuu(u’)2 + 2fufuvu’v’ + fufvv(v’)2 = 0(1 + fu2 + fv2)v’’ + fvfu(u’)2 + 2fvfuvu’v’ + fvfvv(v’)2 = 0Exercise56.Whythenormalvectortothesurfaceatanypointonageodesiccurveshouldbecontainedintheosculatingplaneofthecurveatthatpoint?Giveacleartechnicaljustification.Exercise57.UsingGauss-Bonnettheorem,explainwhyasurfacewithnegativeGaussiancurvaturecannothaveageodesicthatintersectsitself.Exercise58.UsingEq.418↑,provethatallmeridiansofasurfaceofrevolutionaregeodesiccurves.Exercise59.Discussthefollowingstatementinthecontextoftheperceptionofgeodesiccurvesbya2Dinhabitant:“Geodesiccurvesonadevelopablesurfacebecomestraightlineswhenthesurfaceisdevelopedintoaplane”.Exercise60.Giveanexampleofasurfacecurvewhosenormalcurvatureandgeodesiccurvatureareidenticallyzerooverthewholecurve.Exercise61.Whatistherelationbetweenalineofcurvatureonasurfaceandtheprincipaldirectionsatthepointsofthecurve?

Exercise62.Provethatifaplaneandasurfaceareintersectingataconstantanglethentheircurveofintersectionisalineofcurvature.Exercise63.Repeatthepreviousexercisereplacingplanewithsphere.Exercise64.Canalineofcurvatureincludeumbilicalpoints?Discussthisissueconsideringthequestionofallowingmorethantwoprincipaldirectionsatapointornot.Exercise65.Provethatforanysufficientlysmoothsurfaceofrevolution,theparallelsandmeridiansarelinesofcurvature.Exercise66.Showthatforagivennon-umbilicalpointPonasufficientlysmoothsurfacethereisapatchthatcontainsPwherethedirectionsofthecoordinatecurvesatPareprincipaldirections.Exercise67.ProveHilbertlemmausingtheproposalthatthecoordinatecurvesonapatchcancoincidewiththelinesofcurvatureintheneighborhoodofanon-umbilicalpoint.Exercise68.Provethatifthecurveofintersectionoftwosurfacesisalineofcurvatureforonesurfacethenitisalineofcurvaturefortheothersurfacewhenthetwosurfacesareintersectingeachotherataconstantangle.Exercise69.Outlinetheroleofgeodesictorsionincharacterizingthelineofcurvatureemployingamathematicalformulationinthiscontext.Exercise70.Givetwoexamplesforthelineofcurvatureonspecifictypesofsurfacediscussingineachcasewhythedescribedcurveshouldbealineofcurvature.Exercise71.Givetheformulaefortheprincipalcurvatures,κ1andκ2,whentheu1andu2coordinatecurvesofasurfacepatcharelinesofcurvature.Exercise72.Usingtensornotation,statethemathematicalconditionthatshouldbemetbyalineofcurvatureonaspacesurfacewithfullexplanationsofallthesymbolsinvolved.Exercise73.Whichtypesofsurfaceshouldbeexcludedfromthefollowingstatement:“Thelinesofcurvatureonasurfacearerepresentedbyanorthogonalnetonitssphericalimage”?Exercise74.ProvethatonaMongepatchoftheformr(u, v) = (u, v, f(u, v)),thecoordinatecurvesareorthogonalfamilyifffufv = 0identically.Exercise75.Giveamathematicalconditionforadirectiononasurfaceatagivenpointtobeasymptotic.Exercise76.Provethattheasymptoticdirectionsarebisectedbythelinesofcurvature.Exercise77.Givearigoroustechnicaldefinitionofasymptoticline.Exercise78.Whythesecondfundamentalformatapointofasurfaceshouldvanishintheasymptoticdirection?Exercise79.ProveBeltrami-Ennepertheorem(seeEq.428↑andsurroundingtext).Exercise80.Oneofthecharacteristicfeaturesofasymptoticlineisthatthetangentplanetothesurfaceateachpointofthelinecoincideswiththeosculatingplaneofthelineatthatpoint.Why?Exercise81.Showthatonasmoothsurfacewithorthogonalfamiliesofasymptoticlinesthemeancurvatureiszero.Exercise82.Justifythefollowingstatement:“Thenecessaryandsufficientconditionfortheu1andu2coordinatecurvestobeasymptoticlinesisthate = 0identicallyontheu1coordinatecurvesandg = 0identicallyontheu2coordinatecurves”.Exercise83.UsingEq.426↑,provethatthegeneratorsofacircularcylinderareasymptoticlines.Exercise84.AccordingtothetheoremofBeltrami-Enneperwehave:τ2 =  − KwhereτandKstandfortorsionandGaussiancurvature.Doesthismeanthatτisimaginary?Fullyjustifyyouranswer.Exercise85.Theangleθwhichanasymptoticdirectionmakeswiththeprincipaldirectionofκ1isgivenby:tan2θ =  − (κ1 ⁄ κ2)whereκ1andκ2aretheprincipalcurvatures.Derivethisequation.Exercise86.Classifytheasymptoticdirectionsatagivenpointonasurfaceasrealanddistinct,orrealandcoincident,orconjugateimaginaryaccordingtothedeterminantofthesurfacecovariantcurvaturetensoratthatpoint.Exercise87.WhytheclassificationinthepreviousquestioncanalsobebasedontheGaussiancurvatureofthepoint?Exercise88.Theclassificationinthetwopreviousquestionsisrelatedtothenumberofasymptoticdirectionsatelliptic,hyperbolicandparabolicpoints.How?Exercise89.Justifythefollowingstatement:“Astraightlinecontainedinasurfaceisanasymptoticline”.Exercise90.Stateamathematicalconditionfortwodirectionsatagivenpointonasurfacetobeconjugatedirectionsexplainingallthesymbolsused.

Exercise91.What“conjugatefamiliesofcurvesonasurface”means?Exercise92.Showthatathyperbolicandellipticpointseachdirectionhasauniqueconjugatedirection.Exercise93.Showthatonasurfacerepresentedby:r = r1(u) + r2(v)thecoordinatecurvesareconjugatefamiliesofcurves.Exercise94.Whatisthenecessaryandsufficientconditionfortheu1andu2coordinatecurvesonasmoothsurfacetobeconjugatefamilies?

Chapter6SpecialSurfacesTherearemanyclassificationstosurfacesin3Dspacesdependingontheirpropertiesandrelations.Afewoftheseclassificationsarebrieflyinvestigatedinthefollowingsectionsofthischapter.

6.1 PlaneSurfacePlanesaresimple,ruled,connected,elementarysurfaces.Thefollowingstatementsapplytoplanes:1.Allthecoefficientsofthesurfacecurvaturetensorvanishidenticallythroughoutplanesurfaces.2.TheRiemann-Christoffelcurvaturetensorvanishesidenticallyoverplanesurfaces.3.TheGaussiancurvatureKandthemeancurvatureHvanishidenticallyoverplanes.4.Planesareminimalsurfaces(see§6.7↓).5.Allpointsonplanesareflatumbilical.6.Atanypointonaplanesurface,κ1 = κ2 = 0andhenceallthedirectionsareprincipaldirections(orthereisnoprincipaldirection).7.Atanypointonaplanesurface,allthedirectionsareasymptotic.8.Asufficientandnecessaryconditionforasurfacetobeplaneishavinganidenticallyvanishingsurfacecurvaturetensor.9.AsufficientandnecessaryconditionforasurfacetobeisometricwiththeplaneishavinganidenticallyvanishingRiemann-Christoffelcurvaturetensor.ThesameappliesforidenticallyvanishingGaussiancurvature.

6.2 QuadraticSurfaceQuadraticsurfacesaredefinedbythefollowingquadraticequation:(433)Aijxixj + Bixi + C = 0wherei, j = 1, 2, 3,thecoefficientsAijandBiarereal-valuedtensorsofrank-2andrank-1respectivelyandCisarealscalar.Therearemanydegenerateandnon-degeneratetypesofquadraticsurface.However,weconsiderhereonlysixnon-degeneratetypeswhichareprobablythemostcommonlyoccurringindifferentialgeometry.Thesesixtypesare:ellipsoid,hyperboloidofonesheet,hyperboloidoftwosheets,ellipticparaboloid,hyperbolicparaboloid,andquadriccone.Thefirstfiveofthesequadraticsurfaceshavebeendefinedin§1.4.1↑usingparametricforms.Byrigidmotiontransformations,consistingoftranslationandrotationofcoordinatesystem,whosepurposeistoputthecenterofsymmetryorvertexofthesesurfacesattheoriginofcoordinatesandorienttheiraxesandplanesofsymmetrywiththecoordinatelinesandcoordinateplanes,thesetypescanbegiveninthefollowingcanonicalformswhereweassumeaEuclidean3DspacewitharectangularCartesiancoordinatesystem:1.Ellipsoid(Fig.51↓a):(434)(x2 ⁄ a2) + (y2 ⁄ b2) + (z2 ⁄ c2) = 12.Hyperboloidofonesheet(Fig.51↓b):(435)(x2 ⁄ a2) + (y2 ⁄ b2) − (z2 ⁄ c2) = 13.Hyperboloidoftwosheets(Fig.51↓c):(436)(x2 ⁄ a2) − (y2 ⁄ b2) − (z2 ⁄ c2) = 14.Ellipticparaboloid(Fig.51↓d):(437)(x2 ⁄ a2) + (y2 ⁄ b2) − z = 05.Hyperbolicparaboloid(Fig.51↓e):(438)(x2 ⁄ a2) − (y2 ⁄ b2) − z = 06.Quadriccone(Fig.51↓f):(439)(x2 ⁄ a2) + (y2 ⁄ b2) − (z2 ⁄ c2) = 0Wenotethatintheaboveequationsa, b, carerealparameters(seeFootnote33in§8↓),andforconvenienceweusex, y, zforx1, x2, x3respectively.Also,forthefirstthreeofthesesurfacestheoriginofcoordinatesisnotavalidsurfacepoint,asseenfromtheirequations.

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Figure51 Quadraticsurfaces(fromupperrowtolowerrowlefttoright):ellipsoid,hyperboloidofonesheet,hyperboloidoftwosheets,ellipticparaboloid,hyperbolicparaboloid,andquadriccone.

6.3 RuledSurfaceA“ruledsurface”,or“scroll”,isasurfacegeneratedbyacontinuoustranslational-rotationalmotionofastraightlineinspace.Hence,ateachpointofthesurfacethereisastraightlinepassingthroughthepointandlyingentirelyinthesurface.Planes,cones,cylindersandMobiusstrips(Fig.1↑)arecommonexamplesofruledsurface.Theparaboliccylinder(Fig.52↓)isanotherexampleofruledsurface.Thedifferentperspectivesofthegeneratinglinealongitsmovementaredescribedastherulingsofthesurface.Aruledsurfacethatcanbegeneratedbytwodifferentfamiliesoflinesiscalleddoubly-ruledsurface.Examplesofdoubly-ruledsurfacearehyperbolicparaboloids(Fig.53↓)andhyperboloidsofonesheet(Fig.54↓).Atanypointofaregularruledsurface,theGaussiancurvatureisnon-positive(K ≤ 0).

Figure52 Paraboliccylinderasaruledsurface.

Figure53 Hyperbolicparaboloidasadoubly-ruledsurfacewherethegriddemonstratesthetwosetsofstraightlinerulings.

Figure54 Hyperboloidofonesheetasadoubly-ruledsurfacewherethetwoframesdemonstratethetwosetsofstraightlinerulings.Thetangentsurface(see§6.6↓)ofasmoothcurveisaruledsurfacegeneratedbythetangentlineofthecurve.Thetangentplaneisconstantalongabranch,representedbythetangentlineatagivenpoint,ofthetangentsurfaceofacurve.IfPisapointonacurveCwhereChasatangentsurfaceS,thenthetangentplanetoSalongtherulingthatpassesthroughPcoincideswiththeosculatingplaneofCatP.Hence,thetangentsurfacemaybedescribedastheenvelopeoftheosculatingplanesofthecurve.

6.4 DevelopableSurfaceAsdefinedpreviously(see§3.1↑),asurfacethatcanbeflattenedintoaplanewithoutlocaldistortioniscalleddevelopablesurface.AdevelopablesurfacecanalsobedefinedasasurfacethatisisometrictotheEuclideanplane.In3Dmanifolds,alldevelopablesurfacesareruledsurfacesbutnotallruledsurfacesaredevelopablesurfaces.Aruledsurfaceisdevelopableifthetangentplaneisconstantalongeveryrulingofthesurfaceasitisthecasewithconesandcylinders.TheneighborhoodofeachpointonasufficientlysmoothsurfacewithnoflatpointsisdevelopableifftheGaussiancurvaturevanishesidenticallyonthesurface.Thegeneratorsofadevelopablesurfaceandtheirorthogonaltrajectoriesareitslinesofcurvature.Adevelopablesurface,excludingcylinderandcone,isatangentsurfaceofacurvewheretheosculatingplanesofthecurveformthetangentplanesofthesurface.ThecollectionofnormallinestoasurfaceSalongagivencurveConSmakeadevelopablesurfaceiffCisalineofcurvature(see§5.8↑).Intrinsically,anydevelopablesurfaceisequivalent(i.e.havingthesamemetriccharacteristics)toaplaneandhenceanytwodevelopablesurfacesareisometrictoeachother.

6.5 IsometricSurfaceAnisometryisaninjectivemappingfromasurfaceStoasurfaceSwhichpreservesdistances.Ifthemappingpreservesdistancesbutitisnotinjectiveitisdescribedaslocalisometry.Asaconsequenceofpreservingthelengthsinisometricmappings,theanglesandareasarealsopreserved.Examplesofisometricsurfacesarecylinderandconewhicharebothisometrictoplane.Twoisometricsurfaces,suchasacylinderandaconeoreachoneoftheseandaplane,appearidenticaltoa2Dinhabitant.Anydifferencebetweenthetwocanonlybeperceivedbyanexternalobserverresidinginareferenceframeintheenvelopingspace.Accordingly,twoisometricsurfacespossessidenticalfirstfundamentalformsandhenceanydifferencebetweenthem,asviewedextrinsicallyfromtheembeddingspace,isbasedonthedifferencebetweentheirsecondfundamentalforms.Isometryisanequivalencerelationandhenceitisreflective,symmetricandtransitive,thatisforthreesurfacesS1,S2andS3wehave:1.S1 ~ S1.2.S1 ~ S2 ⇔ S2 ~ S1.3.IfS1 ~ S2andS2 ~ S3thenS1 ~ S3.wherethesymbol ~ representsanisometricrelation.IftwosufficientlysmoothsurfaceshaveconstantequalGaussiancurvaturethentheyarelocallyisometric.Themappingrelationbetweenthetwosurfacesthenincludethreeconstantscorrespondingtothethreeindependentcoefficientsofthefirstfundamentalform.Asurfaceofrevolutionisisometrictoitselfininfinitelymanyways,eachofwhichcorrespondstoarotationofthesurfacethroughagivenanglearounditsaxisofsymmetry.Asindicatedbefore,anysurfaceisisometrictotheplaneifftheGaussiancurvature(ortheRiemann-Christoffelcurvaturetensor)vanishesidenticallyonthesurface.

6.6 TangentSurfaceAsstatedpreviously,thetangentsurfaceofaspacecurveisasurfacegeneratedbytheassemblyofallthetangentlinestothecurve.Thetangentlinesofthecurvearecalledthegeneratorsorbranchesofthetangentsurface.Accordingly,theequationofatangentsurfaceSTtoacurveCisgivenby:(440)rT = ri + kTiwhererTisanarbitrarypointonthetangentsurface,riisagivenpointonthecurveC,kisarealvariable( − ∞ < k < ∞),andTiistheunitvectortangenttoCatri.ThetangentsurfaceisgeneratedbyvaryingialongCandkalongthetangentline.Thetangentsurfaceofacurveismadeoftwoparts:onepartcorrespondstok > 0andtheotherpartcorrespondstok < 0wherethecurveisaborderlinebetweenthesetwoparts(seeFig.55↓).Thetwopartsofthetangentsurfacearetangenttoeachotheralongthecurvewhichformsasharpedgebetweenthetwo.Thecurveis,therefore,calledtheedgeofregressionofthesurface.Thetangentplaneisconstantalongabranchofthetangentsurfaceofacurve.Thistangentplaneistheosculatingplaneofthecurveatthepointofcontactofthebranchwiththecurve.Accordingtothedefinitionofinvolute,alltheinvolutesofacurveCearewhollyembeddedinthetangentsurfaceofCe.ThenormaltothetangentsurfaceofaspacecurveCatapointofagivenrulingZisparalleltothebinormallineofCatthepointofcontactofCwithZ.

Figure55 SpacecurveCandthetwopartsofitstangentsurface,shadeddifferentlyatthetopandthebottom.

6.7 MinimalSurfaceAminimalsurfaceisasurfacewhoseareaisminimumcomparedtotheareaofanyothersurfacesharingthesameboundary.Hence,theminimalsurfaceisanextremumwithregardtotheintegralofareaoveritsdomain.Acommonphysicalexampleofaminimalsurfaceisasoapfilmformedbetweentwocoaxialcircularringswhereittakestheminimalsurfaceshapeofacatenoid(Fig.10↑)duetothesurfacetension.Thisproblem,anditsalikeofinvestigationsrelatedtothephysicalrealizationofminimalsurfaces,maybedescribedasthePlateauproblem.Geometricexamplesofminimalsurfaceshapesareplanes,catenoids,helicoids(Fig.11↑)andennepers(Fig.13↑).SincethemeancurvatureHofasurfaceatagivenpointPisameasureoftherateofchangeofareaofthesurfaceelementsintheneighborhoodofP,aminimalsurfaceischaracterizedbyhavinganidenticallyvanishingmeancurvatureandhencetheprincipalcurvaturesateachpointhavethesamemagnitudeandoppositesigns(seeEq.382↑).Aminimalsurfaceisalsocharacterizedbyhavinganorthogonalnetofasymptoticlinesandaconjugatenetofminimallines(seeFootnote34in§8↓).Infact,havinganorthogonalnetofasymptoticlinesisasufficientandnecessaryconditionforhavingzeromeancurvature(referto§5.9↑).Amongsurfacesofrevolution,catenoidistheonlyminimalsurfaceofthistype.

6.8 ExercisesExercise1.Statethreefeatureswhicharespecifictoplanesurfaces.Exercise2.Whyalldirectionsareasymptoticatanypointonaplanesurface?Exercise3.Showthathavinganidenticallyvanishingsurfacecurvaturetensorisanecessaryandsufficientconditionforasurfacetobeplane.Exercise4.ShowthatplaneistheonlyconnectedsurfaceofclassC2whoseallpointsareflat.Exercise5.Givethetensornotationformoftheequationthatdefinesquadraticsurfaces.Exercise6.NamethreetypesofquadraticsurfacegivingtheircanonicalequationinCartesiancoordinates.Exercise7.Asurfaceisrepresentedparametricallyby:r(u, v) = (u + v, u − v, 2uv).ObtainthesurfacerepresentationincanonicalCartesianformandhencedetermineitstype.Exercise8.Makeasimple3DplotofahyperbolicparaboloidshowingtheCartesiancoordinateaxesandindicatingtheparametersofthesurface.Useacomputergraphicpackageifconvenient.Exercise9.Forwhichtypesofquadraticsurfacetheoriginofcoordinatesisnotavalidpointonthesurfaceaccordingtotheircanonicalformsandwhy?Doesthisalsoapplytothenon-canonicalformsofthesesurfaces?Assumingacanonicalform,arethereanylimitingconditionsunderwhichtheorigincanbeincludedinthesesurfaces?Exercise10.Findtheparametricrepresentationofacylindricalsurfacewhoseintersectionwiththexyplaneisgivenby:4x2 + 9y2 = 1andwhosecentralaxisisthez-axis.Whatisthetypeofthissurface?Exercise11.Whatis“ruledsurface”?Whatistheothernamegiventothistypeofsurfaceandwhy?Exercise12.Makesimplesketchesforplane,cone,cylinderandMobiusstripthatdemonstratetheirnatureasruledsurfaces.Exercise13.What“doubly-ruledsurface”means?Giveanexampleofsuchasurfacewithasimplesketch.Exercise14.Provethatthetangentplaneisconstantalongabranchofthetangentsurfaceofaspacecurve.Exercise15.Provethatthehyperbolicparaboloidisadoubly-ruledsurface.Exercise16.Showthatahelixembeddedinacircularcylinderintersectsallthegeneratorsofthecylinderwithconstantangle.Exercise17.Showthatallpointsonthetangentsurfaceofagivenspacecurveareparabolic.Exercise18.Justifythefollowingstatement:“AtanypointofaruledsurfacetheGaussiancurvatureisnon-positive”.Fromthisperspective,discusssingly-anddoubly-ruledsurfaces.Exercise19.Provethatthetangentplanetoacylinderoraconeisconstantalongtheirgenerators.Exercise20.ProvethatifPisapointonacurveCwhereChasatangentsurfaceS,thenthetangentplanetoSalongtherulingthatpassesthroughPcoincideswiththeosculatingplaneofCatP.Exercise21.Define“developablesurface”givingseveralexamplesofthistypeofsurfacewithanexplanationofwhytheyaredevelopablesurfaces.Exercise22.Giveanexampleofaruledsurfacewhichisnotdevelopable.Exercise23.Statetheconditionforaruledsurfacetobedevelopable.Exercise24.Whyanytwodevelopablesurfacesareequivalenttoeachotherbyhavingthesamemetriccharacteristics?Exercise25.Showthatthegeneratorsofdevelopablesurfacesarelinesofcurvature.Exercise26.ShowthatanecessaryandsufficientconditionforaruledsurfacetobedevelopableisthatitsGaussiancurvaturevanishesidentically.Exercise27.Whatis“isometricmapping”?Giveanexampleofsuchamappingbetweentwocommontypesofsurface.Exercise28.Howtwoisometricsurfacesareseenbya2Dinhabitant?Canhedistinguishbetweenthetwoandwhy?Exercise29.Provethatisometryisasymmetricrelation.Exercise30.Demonstratesymbolicallythatisometricmappingisanequivalencerelation.Exercise31.Howtwoisometricsurfacesarecharacterizedintermsoftheirfirstandsecondfundamentalforms?Providedetailedexplanations.

Exercise32.Showthatcatenoidandhelicoidarelocallyisometric.Exercise33.Whyasurfaceofrevolutionisisometrictoitselfininfinitelymanyways?Demonstrateyouranswerbyanexample.Exercise34.Define“tangentsurface”ofaspacecurvedescriptivelyandmathematically.Exercise35.Whatisthedifferencebetweenthe“tangentsurface”ofacurveandthe“tangentplane”ofasurface?Makedetailedcomparisonsbetweenthetwo.Exercise36.Derivetheequationrepresentingthetangentsurfaceofaspacecurverepresentedby:r(t) = (t2, t − 2, t3 + 5).Exercise37.Whythetangentsurfaceofaspacecurveismadeoftwosections?Howthesesectionsmeetonthecurve?Exercise38.Makeasimple3Dsketchofanarbitrarytwistedspacecurveanditstangentsurfaceshowingpartsofitstwosections.Exercise39.ProvethatthecurvemadebytheintersectionofthenormalplaneofacurveCatagivenpointPwiththetangentsurfaceofChasacuspatP.Exercise40.Writedowntheequationrepresentingthetangentsurfaceofaspacecurveexplainingallthesymbolsusedintheequation.Exercise41.Inthecontextoftangentsurfaceofacurve,what“branch”,“generator”,and“edgeofregression”mean?Makeanattempttojustifythesenames.Exercise42.Whatisthemeaningof“minimalsurface”?Givegeometricandphysicalexamplesofthistypeofsurface.Exercise43.Checkifthesurfacerepresentedparametricallyby:r(θ, φ) = (coshθcosφ, coshθsinφ, θ)isminimalornot.Exercise44.Whyminimalsurfacesarecharacterizedbyhavingidenticallyvanishingmeancurvature?Exercise45.WhatistheimplicationofhavingvanishingmeancurvatureHontheprincipalcurvaturesofthesurfaceatthepointswhereHvanishes?Exercise46.Shouldasurfacewithorthogonalfamiliesofasymptoticlinesbeaminimalsurface?Ifso,why?

Chapter7TensorDifferentiationoverSurfacesThefocusofthischapteristhedifferentiationoftensorfieldsoversurfaces,wheregeneralsurfaceandspacecoordinatesaregenerallyassumed.Ingeneral,tensordifferentiation,whethercovariantorabsolute,overa2DsurfacefollowssimilarrulestotherulesthatapplytotensordifferentiationovergeneralnDcurvedspaces.Someoftheserulesare:1.Thesumandproductrulesofdifferentiationapplytocovariantandabsolutedifferentiationasusual.2.Thecovariantandabsolutederivativesoftensorsaretensors.3.Thecovariantandabsolutederivativesofscalarsandinvarianttensorsofhigherranksarethesameastheordinaryderivatives.4.Thecovariantandabsolutederivativeoperatorscommutewiththecontractionofindices.5.Thecovariantandabsolutederivativesofthemetric,Kroneckerandpermutationtensors(andtheirassociatedtensors)vanishidenticallyinanycoordinatesystem,thatis:(441)aαβ|γ = 0aαβ|γ = 0(442)δαβ|γ = 0δαδβω|γ = 0(443)εαβ|γ = 0εαβ|γ = 0wherethesign|representscovariantorabsolutedifferentiationwithrespecttothesurfacecoordinateuγ.Hence,thesetensorsshouldbetreatedlikeconstantsintensordifferentiation.Anexceptiontotheserulesisthecovariantderivativeofthespacebasisvectorsintheircovariantandcontravariantformswhichisidenticallyzero,thatis:(444)Ei;j = ∂jEi − ΓkijEk =  + ΓkijEk − ΓkijEk = 0(445)Ei

;j = ∂jEi + ΓikjEk =  − ΓikjEk + ΓikjEk = 0butthisisnotthecasewiththesurfacebasisvectorsintheircovariantandcontravariantforms,EαandEα,whosecovariantderivativesdonotvanishidentically.Thereasonisthat,duetocurvature,thepartialderivativesofthesurfacebasisvectorsdonotnecessarilylieinthetangentplaneandhencethefollowingrelations:(446)∂jEi =  + ΓkijEk(447)∂jEi =  − ΓikjEk

whicharevalidintheenvelopingspaceandareusedinEqs.444↑and445↑,arenotvalidonthesurfaceanymore.AtagivenpointPonasufficientlysmoothsurfacewithgeodesicsurfacecoordinatesandrectangularCartesianspacecoordinates,thecovariantandabsolutederivativesreducerespectivelytothepartialandtotalderivativesatP.Thecovariantderivativeofthesurfacebasisvectorsissymmetricinitstwoindices,thatis:(448)Eα;β = ∂βEα − ΓγαβEγ = ∂αEβ − ΓγβαEγ = Eβ;αThecovariantderivativeofthesurfacebasisvectors,Eα;β,representsspacevectorswhicharenormaltothesurfacewithnotangentialcomponent.Thecovariantderivativeofadifferentiablerank-1surfacetensorAinitscovariantandcontravariantformswithrespecttoasurfacecoordinateuβisgivenby:(449)Aα;β = (∂Aα ⁄ ∂uβ) − ΓγαβAγ(covariant)(450)Aα

;β = (∂Aα ⁄ ∂uβ) + ΓαγβAγ(contravariant)wheretheChristoffelsymbolsarederivedfromthesurfacemetric.Thecovariantderivativeofadifferentiablerank-2surfacetensorAinitscovariant,contravariantandmixedformswithrespecttoasurfacecoordinateuγisgivenby:(451)Aαβ;γ = (∂Aαβ ⁄ ∂uγ) − ΓδαγAδβ − ΓδβγAαδ(covariant)(452)Aαβ

;γ = (∂Aαβ ⁄ ∂uγ) + ΓαδγAδβ + ΓβδγAαδ(contravariant)(453)Aβ

α;γ = (∂Aβα ⁄ ∂uγ) − ΓδαγAβ

δ + ΓβδγAδα(mixed)

Moregenerally,foradifferentiablesurfacetensorAoftype(m, n),thecovariantderivativewithrespecttoasurfacecoordinateuγisgivenby:(454)

Thecovariantderivativeofaspacetensorwithrespecttoasurfacecoordinateuαisformedbytheinnerproductofthecovariantderivativeofthetensorwithrespecttothespacecoordinatesxkbythetensorxkα.Thismaybeconsideredasaformofthechainruleofdifferentiation.Forexample,thecovariantderivativeofAiwithrespecttouαisgivenby:(455)Ai

;α = Ai;kxkα

ThecovariantderivativewithrespecttoasurfacecoordinateuβofamixedtensorAiα,whichiscontravariantwithrespecttotransformationina

spacecoordinatexiandcovariantwithrespecttotransformationinasurfacecoordinateuα,isgivenby(seeFootnote35in§8↓):(456)Ai

α;β = (∂Aiα ⁄ ∂uβ) + ΓijkAk

α(∂xj ⁄ ∂uβ) − ΓγαβAiγ

wheretheChristoffelsymbolswithLatinandGreekindicesarederivedrespectivelyfromthespaceandsurfacemetrics.Thispatterncanbeeasilygeneralizedtoamixedtensor

oftype(m, n)whichiscontravariantintransformationsofspacecoordinatesxiandcovariantintransformationsofsurfacecoordinatesuα.Forexample,thecovariantderivativeofatensor

withrespecttouγisgivenby:(457)

Theaboverulescanbeextendedfurthertoincludetensorswithspaceandsurfacecontravariantindicesandspaceandsurfacecovariantindices.ForExample,thecovariantderivativeofatensor

withrespecttoasurfacecoordinateuγ,whereiandjarespaceindicesandαandβaresurfaceindices,isgivenby:(458)

Thisexamplecanbeeasilyextendedtothemostgeneralformofatensorwithanycombinationofcovariantandcontravariantspaceandsurfaceindices.ThecovariantderivativeofthesurfacebasisvectorEα,whichintensornotationisdenotedbyxiα,isgivenby:(459)

Fromthelastequation,weconclude:(460)xiα;β = xiβ;αThisisbecausetheChristoffelsymbolsaresymmetricintheirpairedindices(seeEq.66↑)andwehave∂αβxi = ∂βαxiandxjαxkβ = xkβxjα.Thissymmetryhasalsobeenestablishedearlierusingvectornotationforthesurfacebasisvectors(seeEq.448↑).Themixedsecondordercovariantderivativeofthesurfacebasisvectorsisgivenby:(461)xiα;βγ = bαβ;γni + bαβni;γ = bαβ;γni − bαβaδωbδγxiωwherethecovariantderivativeofthesurfacecovariantcurvaturetensorisgiven,asusual,by:(462)bαβ;γ = (∂bαβ ⁄ ∂uγ) − Γδαγbδβ − ΓδβγbαδThecovariantdifferentiationoperatorsinmixedderivativesarenotcommutativeandhenceforacontravariantsurfacevectorAγ,forinstance,wehave:(463)Aγ

;αβ − Aγ;βα = Rγ

δαβAδ

whereRγδαβistheRiemann-Christoffelcurvaturetensorofthesecondkindforthesurface.Similarly,themixedsecondordercovariant

derivativesofthesurfacebasisvectorssatisfythefollowingrelation:(464)xiα;βγ − xiα;γβ = Rδ

αβγxiδThis,infact,isaninstanceofthegeneralrelation:Aj;kl − Aj;lk = Ri

jklAiwhichisfoundintensorcalculustexts.Asdefinedintensorcalculusbooks,theabsoluteorintrinsicderivativeofatensorfieldalongat-parameterizedcurveinannDspacewithrespecttotheparametertistheinnerproductofthecovariantderivativeofthetensorandthetangentvectortothecurve.Thisidenticallyappliestotheabsolutederivativeofcurvescontainedin2Dsurfaces.Consequently,theabsolutederivativeofatensorfieldalongat-parameterizedcurveonasurfacewithrespecttotheparametertfollowssimilarrulestothoseofaspacecurveinageneralnDspace,asstatedintheliteratureoftensorcalculus.Forexample,theabsolutederivativeofadifferentiablesurfacevectorfieldAinitscovariantandcontravariantformswithrespecttotheparametertisgivenby:(465)δAα ⁄ δt = (dAα ⁄ dt) − ΓγαβAγ(duβ ⁄ dt)

(466)δAα ⁄ δt = (dAα ⁄ dt) + ΓαγβAγ(duβ ⁄ dt)wheretheChristoffelsymbolsarederivedfromthesurfacemetric.ItshouldberemarkedthatifAisaspacevectorfielddefinedalongtheabovesurfacecurvethentheaboveformulaewilltakeasimilarformbutwithchangefromsurfacetospacecoordinates,andhencethecurveistreatedasaspacecurve,thatis:(467)δAi ⁄ δt = (dAi ⁄ dt) − ΓjikAj(dxk ⁄ dt)(468)δAi ⁄ δt = (dAi ⁄ dt) + ΓijkAj(dxk ⁄ dt)wheretheChristoffelsymbolsarederivedfromthespacemetric.TheabsolutederivativeofthetensorAi

α,whichisdefinedinthepreviousparagraphs,alongat-parameterizedsurfacecurveisgivenby:(469)δAi

α ⁄ δt = Ai

α;β(duβ ⁄ dt) = [(∂Ai

α ⁄ ∂uβ) + ΓijkAkα(∂xj ⁄ ∂uβ) − ΓγαβAi

γ](duβ ⁄ dt)Thepatternofabsolutedifferentiation,asseenintheaboveexamples,canbeeasilyextendedtoamoregeneraltypeoftensorwithcovariantandcontravariantspaceandsurfaceindices,asdoneforcovariantdifferentiation.ToextendtheideaofgeodesiccoordinatestodealwithmixedtensorsofthetypeAi

α,arectangularCartesiancoordinatesystemoverthespaceandageodesicsystemonthesurfaceareintroducedandhenceatthepolestheabsoluteandcovariantderivativesbecometotalandpartialderivativesrespectively.Asindicatedabove,thecovariantandabsolutederivativesofspaceandsurfacemetric,permutationandKroneckertensorsintheircovariant,contravariantandmixedformsvanishidenticallyandhencetheybehaveasconstantswithrespecttotensordifferentiationwheninvolvedininnerorouterproductoperationswithothertensorsandcommutewiththeseoperators.Similarly,thesurfacecovariantandabsolutederivativesofspacemetrictensor,spaceKroneckertensor,spacepermutationtensorandspacebasisvectorsvanishidentically,thatis:(470)gij|γ = 0gij|γ = 0(471)δij|γ = 0δijkl|γ = 0(472)εijk|γ = 0εijk|γ = 0(473)Ei|γ = 0Ei

|γ = 0wherethesign|representscovariantorabsolutedifferentiationwithrespecttothesurfacecoordinateuγ.Hence,thesespacetensorsareinlieuofconstantswithrespecttosurfacetensordifferentiation.Thenabla∇baseddifferentialoperations,suchasgradientanddivergence,applytospacesurfaceasforanygeneralcurvedspaceandhencetheformulaegivenintheliteratureoftensorcalculusforageneralnDspacecanbeusedwiththesubstitutionofthesurfacecoordinatesandsurfacemetricparameters.Forexample,thedivergenceofasurfacevectorfieldAαisgivenby:(474)∇⋅A = [1 ⁄ √(a)]∂α[√(a)Aα]andtheLaplacianofasurfacescalarfieldfisgivenby:(475)∇2f = [1 ⁄ √(a)]∂α[√(a)aαβ∂βf]whereaαβisthesurfacecontravariantmetrictensor,aisthedeterminantofthesurfacecovariantmetrictensorandα, β = 1, 2.

7.1 ExercisesExercise1.SummarizethemainrulesthatgovernthedifferentiationoftensorfieldsoversurfacesandcomparetheserulestothoseofnDspaces(n > 2).Exercise2.IsthereanyruleoftensordifferentiationthatappliestonDspaces(n > 2)butnottosurfaces?Ifso,whichandwhy?Stateyouranswerwithafullformalexplanation.Exercise3.AtthepointsofasmoothsurfacewithgeodesicsurfacecoordinatesandCartesianspatialcoordinatesofaflatembedding3Dspace,whathappenstothecovariantandabsolutederivativesoftensorfields?Exercise4.Derivethefollowingidentity:Eα;β = Eβ;α.Exercise5.Expresstheidentityinthepreviousexerciseinfulltensornotation.Exercise6.IsEα;βasurfacevectororaspacevector?Discussthepossibledifferentmeaningsoftheseattributes.Exercise7.Explain,indetail,thefollowingequationrelatedtothecovariantderivativeofspacetensorswithrespecttosurfacecoordinates:Ai

;α = Ai;kxkα.

Exercise8.WritedownthemathematicalexpressionforthecovariantderivativeofthetensorAδj

knβwithrespecttothesurfacecoordinateuγwheretheLatinandGreekindicesrepresentspaceandsurfacegeneralcoordinates.Exercise9.Writedownthetensorequationforthecovariantderivativeofthesurfacebasisvectorxmγwithrespecttotheindexβ.Exercise10.CompletethefollowingequationwhichinvolvesthetensorBwheretheindicesrepresentsurfacecoordinates:Bα

;γδ − Bα;δγ = ?

Exercise11.Giveabriefdescriptivedefinitionoftheabsolutedifferentiationofatensorfieldalongacurve.Exercise12.Whatistheothernamegiventotheabsolutedifferentiation?Exercise13.Explainthemathematicalpatternofabsolutedifferentiationoftensorfieldsalongsurfacecurvesillustratingthisbyanexample.Exercise14.Isthepatternofabsolutedifferentiationofsurfacetensorfieldsalongsurfacecurvesidenticaltothepatternofabsolutedifferentiationofspacetensorfieldsalongspacecurves?Ifthereisanydifference,identifyandexplain.Exercise15.Write,inexpandedform,themathematicalequationofthefollowingintrinsicderivative:(δBk

γ ⁄ δt)whereBisatensorandkandγarespaceandsurfaceindices.Exercise16.Whatarethecovariantandabsolutederivativesofspaceandsurfacemetric,permutationandKroneckertensorsintheircovariant,contravariantandmixedforms?Exercise17.Dotheoperatorsofcovariantandabsolutedifferentiationwithrespecttospaceandsurfacecoordinatescommutewiththemetrictensorinvolvedinaninnerorouterproductwithanothertensor?Explainwhy.Exercise18.Explainthefollowingidentitygivingdetaileddefinitionsofallthesymbolsandnotationsinvolved:εijk|γ = 0.Exercise19.Dothenablabaseddifferentialoperationsapplytothesurfacetensorfieldsastothetensorfieldsincurvedspacesofhigherdimensionality?Exercise20.WhatistheLaplacianofadifferentiablecoordinate-dependentsurfacescalarfieldh(i.e∇2h)?Writeinyouranswerthemathematicalequationforthisoperationdefiningallthesymbolsused.Exercise21.ComparetheequationinthepreviousquestionwiththeequationoftheLaplacianofascalarfielddefinedoverageneralnDspace.

References•L.P.Eisenhart.AnIntroductiontoDifferentialGeometry.PrincetonUniversityPress,secondedition,1947.•J.Gallier.GeometricMethodsandApplicationsforComputerScienceandEngineering.Springer,secondedition,2011.•P.Grinfeld.IntroductiontoTensorAnalysisandtheCalculusofMovingSurfaces.Springer,firstedition,2013.•J.H.Heinbockel.IntroductiontoTensorCalculusandContinuumMechanics.1996.•D.C.Kay.Schaum’sOutlineofTheoryandProblemsofTensorCalculus.McGraw-Hill,firstedition,1988.•R.Koch.Mathematics433/533;ClassNotes.UniversityofOregon,2005.•E.Kreyszig.DifferentialGeometry.DoverPublications,Inc.,secondedition,1991.•M.M.Lipschutz.Schaum’sOutlineofDifferentialGeometry.McGraw-Hill,firstedition,1969.•T.Sochi.TensorCalculusMadeSimple.CreateSpace,firstedition,2016.•I.S.Sokolnikoff.TensorAnalysisTheoryandApplications.JohnWiley&Sons,Inc.,firstedition,1951.•B.Spain.TensorCalculus:AConciseCourse.DoverPublications,thirdedition,2003.•J.L.Synge;A.Schild.TensorCalculus.DoverPublications,1978.

AuthorNotes•Allcopyrightsofthisbookareheldbytheauthor.•Thisbook,likeanyotheracademicdocument,isprotectedbythetermsandconditionsoftheuniversallyrecognizedintellectualpropertyrights.Hence,anyquotationoruseofanypartofthebookshouldbeacknowledgedandcitedaccordingtothescholarlyapprovedtraditions.

8 FootnotesFootnote1.ThesurfacecoordinatesmayalsobecalledtheGaussiancoordinates.Footnote2.Themainfocushereisthepropertiesofsurfacesalthoughsomeofthesequalificationscanbeextendedtosurfacecurves.Footnote3.Weareassumingthecurveisembeddedinthexzplane.Footnote4.The“lengthofastraightlinesegment”maybetakenasanaxiomaticconcept.Footnote5.The“areaofapolygonalplanefragment”maybetakenasanaxiomaticconcept.Footnote6.Themeaningof“admissiblecoordinatetransformation”mayvarydependingonthecontext.Wearegenerallyassuminglineartransformations.Footnote7.Thepointsareusuallyassumedtobetotallyconnectedsothatanypointonthecurvecanbereachedfromanyotherpointbypassingthroughothercurvepointsoratleasttheyarepiecewiseconnected.Wealsoconsidermostlyopencurveswithsimpleconnectivityandhencethecurvedoesnotintersectitself.Footnote8.Inthisformula,tisagenericsymbolandhenceitstandsfors.Footnote9.SinceweemployCartesiancoordinatesinaflatspace,ordinaryderivatives(i.e.d ⁄ dsandd ⁄ dt)areusedinthisandthefollowingformulae.Forgeneralcurvilinearcoordinates,theseordinaryderivativesshouldbereplacedbyabsolutederivatives(i.e.δ ⁄ δsandδ ⁄ δt)alongthecurves.Footnote10.Thereisamoretechnicalandrigorousdefinitionofsimplesurfacewhichtheinterestedreadersareadvisedtoseekintheliteratureoftopology.Footnote11.Thisshouldbeunderstoodinaninfinitesimalsenseor,equivalently,asavectorlyinginthetangentplaneofthesurfaceatthegivenpoint,aswillbeseennext.Footnote12.Usingwell-knownidentitiesfromvectoralgebrapluswhatwewillseelaterinthischapter,weobtain:|E1 ×  E2| = √(|E1|2| E2|2 − [ E1⋅ E2]2) = √(a11a22 − [a12]2) = √(a)andhencetheaboveequalityisfullyjustified.Footnote13.Inthistypeofstatements,themeaningisthatwecanfindacoordinatesystemoneachsurfacesuchthattheseconditionsaresatisfied,asindicatedearlyinthebook.Footnote14.Thepurposeofthisdemonstration,whichemploysatypicalsurfacewithinfinitesimaldisplacementofthetangentplane,istoprovideavisualqualitativeimpressionfortheideaofDupinindicatrix.WenotethattheindicatrixisnotnecessarilyrepresentedbyoneofthesecontoursonthesurfaceastheindicatrixisaquadraticequationbasedonthesecondfundamentalformataparticularpointPonthesurface,andhenceitisnotnecessarilysatisfiedbythesurfaceintheextendedneighborhoodofP.Footnote15.Itiscircleifthepointisumbilical(see§4.10↑).Footnote16.WeareconsideringherethepartofthesecurvesintheneighborhoodofthepointPonthesurface.Footnote17.Thismaybedemonstratednon-rigorouslyas:(ds ⁄ du1)(ds ⁄ du1) = (Edu1du1) ⁄ (du1du1) = Esinceontheu1coordinatecurveswehave:IS = (ds)2 = E(du1)2(seeEq.233↑).Hence,du1 ⁄ ds = 1 ⁄ √(E).Footnote18.Thiscanbedemonstratednon-rigorouslyasinthepreviousfootnotebyreplacingdu1withdu2andEwithG.Footnote19.TheplanecontainingnisalsocharacterizedbybeingorthogonaltothetangentplanetothesurfaceatP.Footnote20.Alternatively,wecanuseasingleprincipalnormalvectorforbothnormalsectionswiththeuseoftheprincipalcurvaturesinsteadoftheirabsolutevalues.However,thesignofthesecondtermontherighthandsideoftheaboveequationsshouldbeselectedproperlyasitcanbeplusorminusdependingonthechoiceofndirection.Footnote21.Inbrief,“geodesiccoordinates”herestandsforacoordinatesystemonacoordinatepatchofasurfacewhoseuandvcoordinatecurvefamiliesareorthogonalwithoneofthesefamilies(uorv)beingafamilyofgeodesiccurves.Footnote22.Orthesumdependingontheauthorsalthoughitwillnotbeameananymore.Footnote23.Theseparallelscorrespondtothetwocirclescontactingitstwotangentplanesatthetopandbottomwhichareperpendiculartoitsaxisofsymmetry(seeFig.38↑).Footnote24.Thisisthecommoncase,howeverinsomeexceptionalcasesthesurfaceinthe

neighborhoodofaparabolicpointliesonbothsidesofthetangentplane.Footnote25.Thelattermayalsobecalledsphericalumbilic.Footnote26.Asseen,thesphericalindicatrixmaybeascribedtothevectorortothecurve;themeaningshouldbeobvious.Footnote27.Inthistypeofstatement,whichisfoundincommontextbooksondifferentialgeometry,wemayneedtoaddextrarestrictionssuchasexcludingclosedsurfacesoraddingfurtherconditionslike“intheimmediateneighborhoodofthetwopoints”tomakethestatementapplicabletoalltypesofsurface.Footnote28.Toputitinadifferentway,“geodesiccurves”hastwocommonuses:(a)curveswithidenticallyvanishinggeodesiccurvatureand(b)arcsofoptimallengthconnectingtwopoints,where(b)inasenseisasubsetof(a).Here,“geodesiccurves”isusedinthefirstsense.Footnote29.Therearesomedetailsaboutthisissuethatcouldbeelaboratedwheredifferentconventionsshouldbeconsidered.Themainissueisthat:canthenumberofprincipaldirectionsatagivenpointonasurfaceexceedtwoornot.Hence,someofthefuturematerialsmaynotbebasedonasingleconvention.Footnote30.Theseconditionscanbetakentogetherorseparately.Footnote31.Thediscriminantis:Δ = 4(b12)2 − 4b11b22whilethedeterminantis:b = b11b22 − (b12)2andhenceΔ =  − 4b.Footnote32.Thenotation(δv ⁄ δu)isnotrelatedtothenotationofabsolutederivative(see§7↑).Footnote33.Thesymbolsa, b, cherearedefinedlocallyandhencetheyshouldnotbeconfusedwithsimilarsymbolsusedpreviously;inparticularaandbwhichsymbolizethedeterminantsofthemetricandcurvaturetensors.Footnote34.Minimallinesarecurvesofminimallength.Footnote35.Anexampleofsuchatensorisxiαwhichwasdiscussedearlier,e.g.in§3.3↑.