Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.[i]AlanU.KenningtonDierentialgeometryreconstructedauniedsystematicframeworkFirst edition [work in progress]01234567891290010 010900 100109001001090 010010900100109001001090 010010 90010010900100109001001090010010900100109001001090010010900100100 10900 109001090010900109001090010900109001090010900109001090ABphysicschemistrybiologyn e u r oscienceanthropologylogicmathematics[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.[ii]Mathematics Subject Classication (MSC2000): 5301LibrarycataloguingdataKennington, Alan Ulrich (1953-Differential geometry reconstructed:a unified systematic framework2010 516.3Firstprinting,April2010 [workinprogress]Copyright c ( 2010, Alan U. Kennington.All rights reserved.The author hereby grants permission to print this book draft in A4 format.Printing in all other formats is forbidden.You may not charge any fee for copies of this book draft.This book was typeset by the author with the plain TEX typesetting system.The illustrations in this book were created with MetaPost.This book is available on the Internet: http://www.topology.org/tex/conc/dg.html[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.[iii]PrefaceThis book should be suitable for fourth year university mathematics, physics and engineering students, or foranyone who has already learned dierential geometry but has an uneasy feeling that they may have skimmedover a few too many ne points. The intention here is to replace intuition and hand-waving with a seamless,systematicexposition. However, thisisonlyadenitionsbook, notatheoremsbook. Thereadermustlook elsewhere for serious theorems and serious applications. But understanding denitions is obviously anenormously important part of understanding theorems and applications.Theauthorwrotetherst112pagesofthisbookinearly1992inBonnonhisAtari STcomputer. Afternine years of neglect, he wrote another 310 pages from August 2001 to November 2002. It is still a scruywork in progress scrapbook, but it may be ready for a rst printing some time in 2010. Then there shouldbe one or two printings every year thereafter.Right now, this book looks more like a construction site than a nished building. With some imagination,you may be able to conjure up a vision of the nished work through the scaolding. Material is being addedand rewritten in many chapters and sections simultaneously. The creative process for producing this bookis illustrated in the following diagram. All processes are happening concurrently.booksreadideasbrainwritenotesdesktypeTEX lesworkstationuploadPostScriptlesweb serverdownloadInternetThe current strategy is to rst type in all of my hand-written notes during the ideas capture phase. Thenduringtheconsolidationphase, everythingwill bemadeneat, tidy, comprehensibleandcoherent. Thebookisbeingassembledlikeajigsawpuzzle. Someof thepiecesarettingtogethernicelyalready, butmost pieces are in disorganized heaps. Many pieces are still in the box waiting to be thrown on the table.Sometimes new pieces must be crafted by hand. Its like moving into a new house. First you dump all theboxes on the oor; then you must put everything where it belongs. Inconsistencies, repetition, self-indulgenceand frivolity will be progressively removed. All of the theorems will be proved. All of the exercises will besolved. Formative chaos will yield to serene order. It wont happen overnight, but it will happen.April 2010 Dr. Alan U. KenningtonMelbourne, VictoriaAustraliaDisclaimerThe author of this book disclaims any express or implied guarantee of the tness of this book for any purpose.In no event shall the author of this book be held liable for any direct, indirect, incidental, special, exemplary,or consequential damages (including, but not limited to, procurement of substitute services; loss of use, data,or prots; or business interruption) however caused and on any theory of liability, whether in contract, strictliability,ortort(includingnegligenceorotherwise)arisinginanywayoutoftheuseofthisbook,even ifadvised of the possibility of such damage.BiographyThe author was born in England in 1953 to a German mother and Irish father. The family migrated in 1963to Adelaide, South Australia. The author graduated from the University of Adelaide in 1984 with a Ph.D.inmathematics. HewasatutoratUniversityofMelbournein1984, researchassistantattheAustralianNational University (Canberra) in early 1985, Assistant Professor at University of Kentucky for the 1985/86academic year, and visiting researcher at the University of Heidelberg, Germany, in 1986/87. From 1987 to2007, the author carried out research and development of communications and information technologies inAustralia, Germany and the Netherlands.[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.[iv]Chapters1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Part I. Preliminary topics2. Philosophical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173. Logic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654. Logic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576. Relations and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037. Order and integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2258. Rational and real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479. Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25910. Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28711. Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30512. Ane spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31913. Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32514. Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34915. Topology classes and constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38116. Topological curves, paths and groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40117. Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41718. Dierential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42919. Dieomorphisms in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44720. Measure and integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46321. Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48122. Non-topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48923. Topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49724. Parallelism on topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 529[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.vChaptersPart II. Dierential geometry25. Topological manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54126. Dierentiable manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55127. Tangent bundles on dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . 57128. Tensor bundles and tensor elds on manifolds . . . . . . . . . . . . . . . . . . . . . . . 60129. Higher-order tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61130. Dierentials on manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62331. Higher-order dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63532. Vector eld calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64333. Dierentiable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65334. Dierentiable bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66935. Connections on dierentiable bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . 68136. Ane connections and covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . 69937. Geodesics, convexity and Jacobi elds . . . . . . . . . . . . . . . . . . . . . . . . . . . 71538. Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72339. Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73540. Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73941. Geometry of the 2-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74742. Examples of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76743. Examples of bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77544. Derivations, gradient operators, germs and jets . . . . . . . . . . . . . . . . . . . . . . 78145. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79146. Exercise questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80547. Exercise answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81148. Notations and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82349. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83350. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]vi[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.[vii]Table of contentsChapter 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Layers of structure of dierential geometry . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Topic ow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Chapter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Objectives and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Some minor details of presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Dierences from other dierential geometry texts. . . . . . . . . . . . . . . . . . . . . 111.8 MSC 2000 subject classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9 How to learn mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.10 Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14PartI.Preliminarytopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Chapter 2. Philosophical considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 The bedrock of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Logic, language and tribalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Ontology of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Platos theory of ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 Sets as parameters for socio-mathematical network communications . . . . . . . . . . . 312.6 Sets as parameters for classes of objects. . . . . . . . . . . . . . . . . . . . . . . . . . 362.7 Extraneous properties of set-constructions in denitions . . . . . . . . . . . . . . . . . 382.8 Axioms versus constructions for dening mathematical systems . . . . . . . . . . . . . 402.9 Some general remarks on mathematics and logic . . . . . . . . . . . . . . . . . . . . . 432.10 Dark sets and dark numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.11 Integers and innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.12 Real numbers and innitesimality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Chapter 3. Logic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1 Mathematical logic subject development . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 General comments on logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3 Modelling, meta-modelling and recursive modelling. . . . . . . . . . . . . . . . . . . . 733.4 The universality (or otherwise) of modern logic. . . . . . . . . . . . . . . . . . . . . . 773.5 Logic in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.6 Proposition-store versus world-view ontology for logic . . . . . . . . . . . . . . . . . . 853.7 A proposition-store ontology for logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.8 Undecidable propositions and incomplete information transfer . . . . . . . . . . . . . . 933.9 The semantics of truth and falsity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.10 The semantics of logical negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.11 Proof by contradiction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.12 The moods of logical propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.13 Other remarks on the semantics of logic . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.14 Naive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.viiiChapter 4. Logic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1 Concrete proposition domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2 Logic operations in concrete proposition domains. . . . . . . . . . . . . . . . . . . . . 1164.3 Logical operators and expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.4 Logical expression evaluation and logical argumentation . . . . . . . . . . . . . . . . . 1244.5 Propositional calculus formalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.6 Deduction rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.7 An implication-based propositional calculus. . . . . . . . . . . . . . . . . . . . . . . . 1304.8 Some propositional calculus theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.9 Meta-theorems and the deduction theorem . . . . . . . . . . . . . . . . . . . . . . . 1354.10 Further theorems for the implication operator . . . . . . . . . . . . . . . . . . . . . . 1394.11 Other logical operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.12 Parametrized families of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.13 Logical quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.14 Predicate calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1514.15 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.16 Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Chapter 5. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.1 Zermelo-Fraenkel set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.2 The ZF extension axiom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.3 The ZF empty set, pair, union and power set axioms . . . . . . . . . . . . . . . . . . . 1655.4 The ZF replacement axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.5 The ZF regularity axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.6 The ZF innity axiom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.7 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.8 ZF set theory denitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.9 Axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.10 Axiom of countable choice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1925.11 Zermelo set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.12 Bernays-Godel set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.13 Basic properties of binary set unions and intersections . . . . . . . . . . . . . . . . . . 1945.14 Basic properties of general set unions and intersections. . . . . . . . . . . . . . . . . . 1965.15 Closure of set unions under arbitrary unions . . . . . . . . . . . . . . . . . . . . . . . 1985.16 Specication tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Chapter 6. Relations and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2036.1 Ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2046.2 Cartesian product of a pair of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2066.3 Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.4 Equivalence relations and partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.6 Function set maps and inverse set maps. . . . . . . . . . . . . . . . . . . . . . . . . . 2146.7 Composition of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2166.8 Families of sets and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2176.9 Cartesian products of families of sets and functions. . . . . . . . . . . . . . . . . . . . 2196.10 Partial Cartesian products and identication spaces . . . . . . . . . . . . . . . . . . . 2206.11 Partially dened functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.12 Notations for sets of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.ixChapter 7. Order and integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2257.1 Ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2257.2 Ordinal numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.3 Natural numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.4 Unsigned integer arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2357.5 Signed integers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2357.6 Extended integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377.7 Cartesian products of sequences of sets and functions . . . . . . . . . . . . . . . . . . 2377.8 Choice functions without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . 2387.9 Indicator functions and delta functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.10 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2407.11 Combinations and ordered selections . . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.12 List spaces for general sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2447.13 Reformulation of logic in terms of axiomatic mathematics . . . . . . . . . . . . . . . . 246Chapter 8. Rational and real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2478.1 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2478.2 Extended rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2488.3 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2498.4 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2518.5 Real number tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528.6 Some useful basic real-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2528.7 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Chapter 9. Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2599.1 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2609.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2619.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2649.4 Left transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2679.5 Right transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729.6 Mixed transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2749.7 Figures and invariants of transformation groups . . . . . . . . . . . . . . . . . . . . . 2769.8 Rings and elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2789.9 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2799.10 Associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2829.11 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2839.12 List space for sets with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . 285Chapter 10. Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28710.1 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28710.2 Linear subspaces and basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28910.3 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29110.4 Eigenspaces of linear space endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 29210.5 Linear functionals and dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29310.6 Direct sums of linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29610.7 Quotients of linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29710.8 Inner products and norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29810.9 Groups of linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29910.10 Free linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29910.11 Exact sequences of linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xChapter 11. Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.1 Rectangular matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.2 Component matrices of linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30911.3 Square matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31111.4 Real square matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31311.5 Real symmetric matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31511.6 Real symmetric denite and semi-denite matrices . . . . . . . . . . . . . . . . . . . . 31611.7 Matrix groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316Chapter 12. Ane spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31912.1 Ane spaces discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31912.2 Ane space denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32012.3 Ane transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32312.4 Euclidean spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323Chapter 13. Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32513.1 The meaning of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32513.2 Multilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32813.3 Linear spaces of multilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33113.4 Symmetric and antisymmetric multilinear maps . . . . . . . . . . . . . . . . . . . . . 33213.5 Tensor product metadenition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33313.6 Tensor products of linear spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33513.7 Covariant tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33713.8 Mixed tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33913.9 General tensor algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34013.10 Alternating tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34313.11 Alternating tensor algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34513.12 Tensor products dened via free linear spaces. . . . . . . . . . . . . . . . . . . . . . . 34613.13 Tensor products dened via lists of tensor monomials . . . . . . . . . . . . . . . . . . 347Chapter 14. Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34914.1 Overview of topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35114.2 History and generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35414.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35614.4 Some simple topologies on nite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 35814.5 Interior and closure of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36014.6 Exterior and boundary of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36414.7 Limit points and isolated points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36814.8 Some simple topologies on countably innite sets . . . . . . . . . . . . . . . . . . . . . 36914.9 Generation of topologies from collections of sets . . . . . . . . . . . . . . . . . . . . . 37214.10 The standard topology for the real numbers . . . . . . . . . . . . . . . . . . . . . . . 37414.11 Open bases and open subbases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37414.12 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37614.13 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xiChapter 15. Topology classes and constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38115.1 Product and quotient topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38115.2 Separation classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38315.3 Separation and disconnection of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 38715.4 Connectivity classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38815.5 Denition of continuity of functions using connectivity . . . . . . . . . . . . . . . . . . 39215.6 Open bases, countability classes and separability . . . . . . . . . . . . . . . . . . . . . 39415.7 Compactness classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39515.8 Topological properties of real number intervals . . . . . . . . . . . . . . . . . . . . . . 39715.9 Topological dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39815.10 Set union topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39815.11 Topological identication spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399Chapter 16. Topological curves, paths and groups . . . . . . . . . . . . . . . . . . . . . . . . . . 40116.1 Curve and path terminology and denition options . . . . . . . . . . . . . . . . . . . . 40116.2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40416.3 Path-equivalence relations for curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 40616.4 Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40816.5 Convex curvilinear interpolation in ane spaces . . . . . . . . . . . . . . . . . . . . . 41016.6 Algebraic topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41216.7 Topological groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41316.8 Topological transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41316.9 Topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41416.10 Network topology and continuous paths in networks . . . . . . . . . . . . . . . . . . . 415Chapter 17. Metric spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41717.1 Distance functions and balls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41817.2 Set distance and set diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42017.3 The topology induced by a metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42117.4 Continuous functions in metric spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 42517.5 Rectiable sets, curves and paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426Chapter 18. Dierential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42918.1 Innitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42918.2 Dierentiation for one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43018.3 Unidirectional dierentiability of real-to-real functions . . . . . . . . . . . . . . . . . . 43518.4 Higher-order derivatives for real-to-real functions . . . . . . . . . . . . . . . . . . . . . 43618.5 Dierentiation for several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43918.6 Higher-order derivatives for several variables . . . . . . . . . . . . . . . . . . . . . . . 44318.7 Some dierentiability-based function spaces. . . . . . . . . . . . . . . . . . . . . . . . 44418.8 Dierentiation for abstract linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . 44418.9 H older continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445Chapter 19. Dieomorphisms in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . 44719.1 Tangent vectors and dieomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 44719.2 Dierentials and dieomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44919.3 Second-level tangent vectors and dieomorphisms . . . . . . . . . . . . . . . . . . . . 45419.4 Dieomorphism pseudogroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45719.5 Second-order dierential operators and dieomorphisms . . . . . . . . . . . . . . . . . 45919.6 Directionally dierentiable homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . 461[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xiiChapter 20. Measure and integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46320.1 Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46320.2 Lebesgue integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46420.3 Rectangular Stokes theorem in two dimensions . . . . . . . . . . . . . . . . . . . . . . 46520.4 Rectangular Stokes theorem in three dimensions . . . . . . . . . . . . . . . . . . . . . 46720.5 Dierential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46820.6 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47020.7 Exterior dierentiation using Lie derivatives . . . . . . . . . . . . . . . . . . . . . . . 47320.8 Geometric measure theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47420.9 Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47420.10 Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47420.11 Some integrability-based function spaces . . . . . . . . . . . . . . . . . . . . . . . . . 47520.12 Logarithmic and exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . 47520.13 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477Chapter 21. Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48121.1 Ordinary dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48321.2 Systems of linear second-order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 48521.3 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48521.4 Initial value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48521.5 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48621.6 ODEs for dening exponential and trigonometric functions. . . . . . . . . . . . . . . . 48621.7 Taylor series and exponentials of matrices . . . . . . . . . . . . . . . . . . . . . . . . 487Chapter 22. Non-topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48922.1 Non-topological brations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48922.2 Parallelism for non-topological brations . . . . . . . . . . . . . . . . . . . . . . . . . 49122.3 Non-topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49322.4 Finite transformation groups as bre bundles . . . . . . . . . . . . . . . . . . . . . . . 493Chapter 23. Topological bre bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49723.1 History, motivation and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49723.2 Topological brations with intrinsic bre spaces . . . . . . . . . . . . . . . . . . . . . 50023.3 Topological brations and bre atlases . . . . . . . . . . . . . . . . . . . . . . . . . . 50223.4 Fibration identication spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50623.5 Structure groups discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50723.6 Topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50823.7 Fibre bundle homomorphisms, isomorphisms and products . . . . . . . . . . . . . . . . 51223.8 Structure-preserving bre set maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51423.9 Topological principal bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51923.10 Associated topological bre bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52123.11 Construction of associated topological bre bundles . . . . . . . . . . . . . . . . . . . 52423.12 Construction of associated topological bre bundles via orbit spaces . . . . . . . . . . . 525Chapter 24. Parallelism on topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . 52924.1 Parallelism path classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52924.2 Pathwise parallelism on topological bre bundles . . . . . . . . . . . . . . . . . . . . . 53124.3 Associated parallelism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53424.4 Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xiiiPartII.Dierentialgeometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539Chapter 25. Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54125.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54125.2 Euclidean and locally Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 54325.3 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54425.4 Charts and atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54525.5 Topological manifold constructions, attributes and relations . . . . . . . . . . . . . . . 54825.6 Topological identication spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549Chapter 26. Dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55126.1 Overview of dierentiable structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55226.2 Terminology and denition choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55326.3 Dierentiable manifold atlases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55426.4 Some standard dierentiable manifold atlases. . . . . . . . . . . . . . . . . . . . . . . 55626.5 Some basic denitions for dierentiable manifolds . . . . . . . . . . . . . . . . . . . . 55626.6 Dierentiable real-valued functions on dierentiable manifolds . . . . . . . . . . . . . . 55826.7 Dierentiable curves and paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56026.8 Dierentiable families of dierentiable transformations . . . . . . . . . . . . . . . . . . 56126.9 Dierentiable maps between dierentiable manifolds . . . . . . . . . . . . . . . . . . . 56226.10 Analytic manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56426.11 Unidirectionally dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 56426.12 Lipschitz manifolds and rectiable curves . . . . . . . . . . . . . . . . . . . . . . . . . 56526.13 Dierentiable brations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56726.14 Tangent space building principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569Chapter 27. Tangent bundles on dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . 57127.1 Styles of representation of tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . 57327.2 Tangent bundle metadenition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57727.3 Tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58027.4 Computational tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58127.5 Tangent operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58227.6 Tagged tangent operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58527.7 Pointwise tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58527.8 Tangent bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58727.9 Tangent operator bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59027.10 The tangent bundle of a tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . 59027.11 Horizontal components and drop functions . . . . . . . . . . . . . . . . . . . . . . . . 59427.12 Tangent frames and coordinate basis vectors . . . . . . . . . . . . . . . . . . . . . . . 59627.13 Tangent space constructions, attributes and relations. . . . . . . . . . . . . . . . . . . 59827.14 Unidirectional tangent bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59827.15 Distributions as representations of tangent bundles . . . . . . . . . . . . . . . . . . . . 59927.16 Tangent bundles on innite-dimensional manifolds . . . . . . . . . . . . . . . . . . . . 600[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xivChapter 28. Tensor bundles and tensor elds on manifolds . . . . . . . . . . . . . . . . . . . . . . 60128.1 Contravariant tensors and tensor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 60228.2 Cotangent vectors and cotangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . 60228.3 Covariant and mixed tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60428.4 Double tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60528.5 Vector elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60628.6 Tangent operator elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60728.7 Tensor elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60828.8 Vector elds and tensor elds along curves . . . . . . . . . . . . . . . . . . . . . . . . 60828.9 Dierential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609Chapter 29. Higher-order tangent vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61129.1 Higher-order tangent operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61329.2 Tensorization coecients for second-order tangent operators . . . . . . . . . . . . . . . 61529.3 Higher-order tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61729.4 Higher-order tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62029.5 Drop functions for second-level tangent vectors . . . . . . . . . . . . . . . . . . . . . . 62029.6 Elliptic second-order operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62129.7 Higher-order vector elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62129.8 Higher-order vector elds for families of curves . . . . . . . . . . . . . . . . . . . . . . 622Chapter 30. Dierentials on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62330.1 Pointwise dierentials versus induced maps . . . . . . . . . . . . . . . . . . . . . . . . 62330.2 The dierential of a real-valued function . . . . . . . . . . . . . . . . . . . . . . . . . 62530.3 The dierential of a dierentiable map . . . . . . . . . . . . . . . . . . . . . . . . . . 62730.4 The dierential of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63130.5 One-parameter transformation families and vector elds . . . . . . . . . . . . . . . . . 633Chapter 31. Higher-order dierentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63531.1 Higher-order dierentials of a real-valued function . . . . . . . . . . . . . . . . . . . . 63531.2 Higher-order dierentials of a dierentiable map . . . . . . . . . . . . . . . . . . . . . 63531.3 Higher-order dierentials of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63631.4 Higher-order dierentials of curve families . . . . . . . . . . . . . . . . . . . . . . . . 63831.5 Dierentials of real-valued functions for higher-order operators. . . . . . . . . . . . . . 63931.6 Hessian operators at critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64031.7 Dierentials of dierentiable maps for higher-order operators . . . . . . . . . . . . . . . 64031.8 Dierentials of curves for higher-order operators . . . . . . . . . . . . . . . . . . . . . 642Chapter 32. Vector eld calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64332.1 Naive vector eld derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64332.2 The Poisson bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64532.3 Vector eld derivatives for curve families . . . . . . . . . . . . . . . . . . . . . . . . . 64732.4 Lie derivatives of vector elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64732.5 Lie derivatives of tensor elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65232.6 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xvChapter 33. Dierentiable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65333.1 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65433.2 Hilberts fth problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65533.3 Left invariant vector elds on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . 65633.4 Right invariant vector elds on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 66033.5 The Lie algebra of a Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66233.6 Dieomorphism groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66333.7 Lie transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66433.8 Innitesimal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666Chapter 34. Dierentiable bre bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66934.1 Dierentiable bre bundles with non-Lie structure group. . . . . . . . . . . . . . . . . 67034.2 Dierentiable bre bundles with Lie structure group . . . . . . . . . . . . . . . . . . . 67034.3 Vector elds on dierentiable bre bundles . . . . . . . . . . . . . . . . . . . . . . . . 67134.4 Dierentiable principal bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 67334.5 Vector elds on dierentiable principal bre bundles . . . . . . . . . . . . . . . . . . . 67434.6 Associated dierentiable bre bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . 67434.7 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67734.8 Tangent bundles of dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . 67734.9 Tangent frame bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678Chapter 35. Connections on dierentiable bre bundles . . . . . . . . . . . . . . . . . . . . . . . 68135.1 Naming, history and choice of denitions . . . . . . . . . . . . . . . . . . . . . . . . . 68235.2 Dierentiation of parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68435.3 Horizontal lift functions for ordinary bre bundles . . . . . . . . . . . . . . . . . . . . 68635.4 Curvature of connections on ordinary bre bundles . . . . . . . . . . . . . . . . . . . . 68935.5 Horizontal lift functions for principal bre bundles . . . . . . . . . . . . . . . . . . . . 69035.6 Connection forms for PFB connections . . . . . . . . . . . . . . . . . . . . . . . . . . 69335.7 Covariant derivatives for general connections . . . . . . . . . . . . . . . . . . . . . . . 69535.8 Parallel displacement for PFB connections . . . . . . . . . . . . . . . . . . . . . . . . 69535.9 Alternative denitions for general connections . . . . . . . . . . . . . . . . . . . . . . 696Chapter 36. Ane connections and covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . 69936.1 Concepts, history and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70036.2 Overview of ane connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70136.3 Motivation for dening connections on manifolds . . . . . . . . . . . . . . . . . . . . . 70236.4 Ane connections on tangent bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 70436.5 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70436.6 Hessian operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70736.7 Elliptic second-order operator elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70836.8 Curvature and torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70936.9 Ane connections on principal bre bundles . . . . . . . . . . . . . . . . . . . . . . . 71136.10 Coecients of ane connections on principal bre bundles. . . . . . . . . . . . . . . . 71136.11 Connections for Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 713[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xviChapter 37. Geodesics, convexity and Jacobi elds . . . . . . . . . . . . . . . . . . . . . . . . . . 71537.1 Covariant derivatives of vector elds along curves . . . . . . . . . . . . . . . . . . . . 71537.2 Geodesic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71637.3 Jacobi elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71737.4 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71737.5 Convex combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71837.6 Convex curvilinear interpolation in ane manifolds . . . . . . . . . . . . . . . . . . . 71837.7 Families of geodesic interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71937.8 Exponential maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72037.9 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720Chapter 38. Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72338.1 Historical notes on Riemannian geometry. . . . . . . . . . . . . . . . . . . . . . . . . 72438.2 Overview of Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72538.3 The Riemannian metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72738.4 The point-to-point distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . 72838.5 The Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73038.6 Curvature tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73238.7 Dierential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73238.8 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73338.9 Embedded Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73338.10 Information geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733Chapter 39. Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73539.1 Overview of pseudo-Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . 73539.2 The pseudo-Riemannian metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73639.3 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73739.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73739.5 Global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738Chapter 40. Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73940.1 History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74040.2 Dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74040.3 Manifolds with ane connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74040.4 Equations of geodesic variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74140.5 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74340.6 Pseudo-Riemannian manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74540.7 Submanifolds of Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xviiChapter 41. Geometry of the 2-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74741.1 Terrestrial coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74741.2 Tensor calculus in terrestrial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 74941.3 Metric tensor calculation from the distance function . . . . . . . . . . . . . . . . . . . 75141.4 The principal bre bundle in terrestrial coordinates . . . . . . . . . . . . . . . . . . . 75241.5 The Riemannian connection in terrestrial coordinates . . . . . . . . . . . . . . . . . . 75341.6 Coordinates for polar exponential maps. . . . . . . . . . . . . . . . . . . . . . . . . . 75541.7 The global tangent bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75741.8 Isometries ofS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75841.9 Geodesic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76041.10 Anely parametrized geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76241.11 Convex sets and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76341.12 Normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76341.13 Jacobi elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76341.14 Circles on the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76341.15 Calculation of the hours of daylight. . . . . . . . . . . . . . . . . . . . . . . . . . . 76441.16 Some standard map projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76441.17 Projection of a sphere onto a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764Chapter 42. Examples of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76742.1 Topological space examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76742.2 Euclidean spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76842.3 Non-Hausdor locally Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 76942.4 H older-continuous manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76942.5 Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77142.6 General sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77142.7 Conical coordinates for Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . 77242.8 Hyperboloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77242.9 Tractrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77242.10 Analysis on Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773Chapter 43. Examples of bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77543.1 Euclidean bre bundles on Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . 77543.2 The Mobius strip as a bre bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77643.3 The Mobius strip bre bundle onS1. . . . . . . . . . . . . . . . . . . . . . . . . . . 778Chapter 44. Derivations, gradient operators, germs and jets . . . . . . . . . . . . . . . . . . . . . 78144.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78244.2 Some elementary examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78244.3 Further elementary examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78544.4 Spaces of dierentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78544.5 Spaces of smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78744.6 The space of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78944.7 The Holder spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78944.8 Further topics on derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78944.9 Germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78944.10 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790Chapter 45. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79145.1 Chronology of mathematicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79145.2 Origins of words and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79645.3 Etymology of ane spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79845.4 Logical language in ancient literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 801[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.xviiiChapter 46. Exercise questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80546.1 Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80546.2 Sets, relations and functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80646.3 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80746.4 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80746.5 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80746.6 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80846.7 Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80846.8 Topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80846.9 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80946.10 Dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809Chapter 47. Exercise answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81147.1 Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81147.2 Sets, relations and functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81347.3 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81747.4 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81747.5 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81847.6 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81847.7 Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81947.8 Topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81947.9 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82047.10 Dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821Chapter 48. Notations and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82348.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82348.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831Chapter 49. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83349.1 Dierential geometry introductory texts . . . . . . . . . . . . . . . . . . . . . . . . . 83349.2 Other dierential geometry references. . . . . . . . . . . . . . . . . . . . . . . . . . . 83549.3 Other mathematics references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83649.4 Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83849.5 Logic and set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83849.6 Anthropology and linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83849.7 Philosophy and ancient history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83949.8 History of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83949.9 Other references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83949.10 Comments on other peoples books . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840Chapter 50. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ][1]Chapter1Introduction1.1 Layers of structure of dierential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Topic ow diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Chapter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Objectives and motivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Some minor details of presentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Dierences from other dierential geometry texts . . . . . . . . . . . . . . . . . . . . . . . 111.8 MSC 2000 subject classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9 How to learn mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.10 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14This book is not Dierential Geometry Made Easy. Dierential geometry is not easy. If you think its easy,you havent understood it! Attempts to make it seem easy give the reader only a supercial understanding.The best that can be hoped for realistically is a systematic, self-consistent presentation of topics so that theideas can be assimilated by the reader without any more pain and confusion than absolutely necessary. Thisbook aims to be Dierential Geometry Made Crystal Clear, but enlightenment requires eort.This is a denitions book, not a theorems book. Denitions introduce you to things and tell you their names.Theorems tell you properties and relations of things. Most mathematical texts give denitions so that theycanpresenttheirtheorems. Inthisbook, theoremsaregivenonlywhenrequiredforthepresentationofdenitions. If the reader can understand the denitions in the DG literature, that is a good starting pointforunderstandingthetheorems. Tobemeaningful, manydenitionsdorequireexistence, uniquenessorregularity proofs. So some basic theorems are unavoidable. Also a few theorems are given here to motivatedenitions or to clarify the relations between them.Before studying dierential geometry, the reader should have some prior familiarity with set theory, grouptheory,linear algebra,topology,measuretheory and partial dierential equations. Theseprerequisites arepresented in preliminary chapters in this book, but it is preferable to have studied these topics beforehand.Dierential geometry is the geometry of manifolds. Coordinate charts are the principal dening characteristicof dierential geometry, not anembarrassingnuisance. Thereforecoordinatecharts areconstantlyandunashamedly in the foreground in this book. Some authors dont like coordinates. They can be hidden butnot removed. (The coordinates, that is.)The central concepts of dierential geometry are coordinate charts, tangent vectors, the exterior derivative,pathwise parallelism,curvature and metric tensors. The aim of this book is to give the reader a condentunderstanding of these concepts and the relations between them. The presentation strategy is to stratify allDG concepts according to structural layers. This book will hopefully ll the role of an illustrated dictionary.It does not try to be a comprehensive encyclopedia. It presents only the dramatis personae, not the completeworks of Shakespeare.The author may use this book as a resource for the creation of other books. When this full version has beenreleased, the author may write a half-length version which omits the less popular technicalities. The shorterversion may be titled Dierential Geometry Made Easy. It might sell a lot of copies!AlanU.Kennington, Dierential geometryreconstructed: auniedsystematicframework.Copyright 2010, AlanU. Kennington. Allrightsreserved. YoumayprintthisbookdraftinA4format.Printinginallotherformatsisforbidden. Youmaynotchargeanyfeeforcopiesofthisbookdraft.[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.2 1. Introduction1.1. LayersofstructureofdierentialgeometryThefollowingtablesummarizestheprogressivebuild-upofthelayersofstructureofdierentialgeometryin the chapters of this book.layer main concept structure chapters0 point-set layer points set of points with no topological structure 5131 topological layer connectivity topological space: open neighbourhoods 1417, 23252 dierential layer vectors atlas of dierentiable charts; tangent bundle 26343 connection layer parallelism ane connection on the tangent bundle 35374 metric layer distance Riemannian metric tensor eld 3839The following table shows which levels of structure are required by some important concepts. For example,geodesic curves are well dened if an ane connection or Riemannian metric is specied, but not if you onlyhave a dierentiable structure.concepts structural layers where concepts are meaningfulpoint topology dierentiable ane riemannianset structure connection metric0 cardinality of sets yes yes yes yes yes1 boundaries of sets yes yes yes yesconnectivity of sets yes yes yes yescontinuity of functions yes yes yes yes2 tangent vectors yes yes yesdierentials of functions yes yes yestensor bundles yes yes yesdierential forms yes yes yesvector eld algebra yes yes yesLie derivatives yes yes yesexterior derivative yes yes yesStokes theorem yes yes yes3 parallel transport yes yescovariant derivatives yes yesgeodesic curves yes yesgeodesic coordinates yes yesconvex sets and functions yes yesRiemann curvature yes yesRicci curvature yes yes4 angle between vectors yeslength of vector yesdistance between points yesnormal coordinates yessectional curvature yesscalar curvature yesEinstein curvature tensor yesLaplace-Beltrami operator yesThe specication of any structural layer uniquely determines the lower layer structures but not the higherlayer structures. When only a point set is specied, there are many possible choices for the topology. Whenonly the topology is specied, there are many choices for the dierentiable structure. On a given dierentiablestructure, many choices of connection are possible. But a Riemannian metric uniquely determines the aneconnection, which uniquely determines the dierentiable structure, and so forth.The higher layers are optional. You only need to provide the layers of structure which are required by theconcepts you wish to use. More structure gives you more concepts. It is noteworthy that so many conceptsare well dened in the absence of a metric, and even in the absence of a connection.[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.1.2. Topic ow diagram 31.2. TopicowdiagramChapters2to24presentpreliminarytopicsforreferenceinlaterchapters. Thesetopicsareintheearlychapters so that later chapters are not cluttered by the interpolation of prerequisites. The preliminary topicsinclude logic, sets, functions, order and numbers (Chapters 38), algebra (Chapter 9), linear algebra (Chap-ters 1012), tensor algebra (Chapter 13), topology (Chapters 1417),calculus (Chapters 1821) and bre bundles (Chapters 2224).Dierential geometry begins with topological manifolds (Chapter 25),followed by dierentiable manifolds (Chapter 26), connections (Chap-ters 3537) and Riemannian metric spaces (Chapters 3839).This book tries to disentangle which concepts and theorems belong tofourlevelsofstructure: topologicalstructure(Chapter25), dieren-tiable structure (Chapters 2632), ane connection structure (Chap-ters 3537),andRiemannianmetricstructure(Chapters 3839). Lie(dierentiable)groups(Chapter33)anddierentiablebrebundles(Chapter 34) may be considered as preliminary topics, but like tensoralgebra (Chapter 13) and topological bre bundles (Chapters 2324),they may be regarded as core topics of dierential geometry.Laterchaptersdeal withtensorcalculus(Chapter40), the2-sphere(Chapter 41), example geometries and bre bundles (Chapters 4243),and alternative tangent space denitions (Chapter 44).setsandnumbersalgebratopology linearspacestensoralgebratopologicalmanifoldstopologicalbrebundlesdierentiablemanifoldsLiegroupsdierentiablebrebundlesaneconnectionsRiemannianmanifoldspseudo-RiemannianmanifoldscalculusThetopicowdiagramshows theprogressivebuild-upof algebraicstructure from sets and numbers to tensor algebra. Then topol-ogy and calculus are combined with tensor algebra to dene dierentiable manifolds. Adding topo-logical bre bundles to this yields dierentiable bre bundles on which ane connections are dened.Addingametricorpseudo-metricleadstoRiemannianorpseudo-Riemannianmanifolds. Algebraicandanalytical structure are thus developed in two intermingled streams. This modern approach to dierentialgeometry is expressed in the raried language of bre bundles. Ane connections may be dened directlyon dierentiable manifolds, bypassing bre bundles as suggested by the dashed arrow, like in the olden days.1.3. Chaptergroups1.3.1Remark: The page counts for the chapter groups are as follows.pages chapter group chapters16 introduction 1524 PartI:preliminarytopics 22494 philosophy, semantics 23148 logic, set theory, numbers 4890 algebra 91380 topology 141760 calculus 182152 topological bre bundles 2224250 PartII:dierentialgeometry 254410 topological manifolds 25102 dierentiable manifolds 263228 Lie groups, dierentiable bre bundles 333442 connections 353724 Riemannian metric, tensor calculus 384034 examples 414310 derivations 4450 appendices 454945 index 50[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.4 1. Introduction1.3.2Remark: The chapters of this book fall more or less naturally into the following groups.Chapter 1 is a general introduction. This may be safely ignored apart from Section 1.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1PartI.Preliminarytopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Chapters 2 and 3 discuss philosophy and semantics. These are the most annoying chapters of the book.2. Philosophical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173. Logic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Chapters 4 to 8 present logic, sets, relations, functions, order and numbers.4. Logic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576. Relations and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037. Order and integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2258. Rational and real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247Chapters 9 to 13 introduce algebra, especially linear and multilinear (tensor) algebra.9. Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25910. Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28711. Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30512. Ane spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31913. Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Chapters 14 to 17 introduce topology. Metric spaces are a particular kind of topological space.14. Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34915. Topology classes and constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38116. Topological curves, paths and groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40117. Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Chapters18to21introduceanalyticaltopics,namelythedierentialcalculusandintegralcalculus. Thisprovides a break from topology before returning to it in the bre bundle chapters.18. Dierential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42919. Dieomorphisms in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44720. Measure and integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46321. Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Chapters 22 to 24 introduce bre bundles which have topology but no dierentiable structure.22. Non-topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48923. Topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49724. Parallelism on topological bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 529PartII.Dierentialgeometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539Chapter 25 introduces topological manifolds. This is layer 1 in the ve-layer DG structure model.25. Topological manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541Chapters 26 to 32 add dierentiable structure (i.e. charts) to manifolds. This commences layer 2.26. Dierentiable manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55127. Tangent bundles on dierentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . 57128. Tensor bundles and tensor elds on manifolds . . . . . . . . . . . . . . . . . . . . . . . 60129. Higher-order tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61130. Dierentials on manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62331. Higher-order dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63532. Vector eld calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643Chapters 33 and 34 introduce Lie (i.e. dierentiable) groups. Lie groups are required for the formal denitionof dierentiable bre bundles, which are required for dening general connections.33. Dierentiable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65334. Dierentiable bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.1.4. Objectives and motivations 5Chapters 35 to 37 introduce connections (i.e. dierentiable parallelism) on manifolds. This is layer 3. Con-nections are required for concepts such as covariant derivatives, geodesics, convexity and curvature.35. Connections on dierentiable bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . 68136. Ane connections and covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . 69937. Geodesics, convexity and Jacobi elds . . . . . . . . . . . . . . . . . . . . . . . . . . . 715Chapters 38 to 40 introduce Riemannian and pseudo-Riemannian metrics. This is layer 4. Such metrics arerequired for general relativity. Tensor calculus is a notational system for practical DG calculations.38. Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72339. Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73540. Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739Chapters41to43presentnumerousexamplesof manifoldsandbrebundles. Aparticularlyuseful andfamiliar manifold is the 2-sphereS2.41. Geometry of the 2-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74742. Examples of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76743. Examples of bre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775Chapter44isanotveryuseful setof notesonderivations, germsandjets, whichprovidesomeobscurerepresentations of tangent spaces.44. Derivations, gradient operators, germs and jets . . . . . . . . . . . . . . . . . . . . . . 781Chapters 45 to 50 are appendices to the main part of the book.45. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79146. Exercise questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80547. Exercise answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81148. Notations and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82349. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83350. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8411.4. Objectivesandmotivations1.4.1Remark: Between 1986 and 1991, the author was trying to generalize some geometric properties ofsolutions of second-order boundary and initial value problems from at space to dierentiable manifolds. (Inparticular,theauthorneeded estimates for paralleltransportofsecond-orderpartialdierential operatorsalong geodesics in terms of bounds on curvature.)More importantly, it seemed a terrible shame that mathematicians had developed such a deep and compre-hensive corpus of results for partial dierential equations in at space, particularly for boundary and initialvalue problems, whereas according to cosmologists, the universe is no longer at, in which case a vast swatheof the PDE corpus must surely be null and void, being inapplicable to curved space. Many of the techniquesof PDE theory are in fact very much dependent on the special properties of Euclidean space.Theve-layerstructural organization ofdierential geometry inSection 1.1is a directconsequenceof thedesireto minimizethe requirements placed byDG on PDE so that the maximumextent of generalizationto curved spaces will be facilitated. The requirements minimization objective is considered in choosing alldenitions in this book.For the task of converting PDE concepts and techniques to curved space, the author could not nd dierentialgeometry texts which met the high standards of systematic development and logical rigour of the best analysistexts. The more he read, the more confusing the subject became because of the multitude of contradictorydenitions and formalisms.The origins and motivations of fundamental DG concepts are largely submerged under a century of continuousredenitionandrearrangement. Thedierentialgeometryliteratureisplaguedbyaplethoraofmutuallyincomprehensibleformalismsandnotations. Writingthisbookhasbeenlikecreatingamapoftheworldfrom a hundred regional maps which use dierent coordinate systems and languages for locating and naminggeographical features.1.4.2Remark: Long after initially writing the comments in Remark 1.4.1 regarding a multitude of contra-dictory denitions and formalisms and a plethora of mutually incomprehensible formalisms and notations,[ www.topology.org/tex/conc/dg.html] [ draft: UTC201048Thursday09:22 ]Differential geometry reconstructed. Copyright (C) 2010, Alan U. Kennington. All Rights Reserved. The author hereby grants permission to print this book draft in A4 format. Printing in all other formats is forbidden. You may not charge any fee for copies of this book draft.6 1. Introductionthe author acquired a copy of Michael Spivaks 5-volume DG book. The rst two paragraphs of the prefaceto the 1970 edition ([43], page ix) contain eerily similar comments.[. . . ] no one denies that modern denitions are clear, elegant, and precise; its just that its impossibletocomprehendhowanyoneeverthoughtof them. Andevenafteronedoesmasteramoderntreatment of dierential geometry, other modern treatments often appear simply to be about totallydierent subjects.Since 1970, it seems little has changed. If anything, the literature is now even more confusing.1.4.3Remark: The initial strategy of this book was to stitch together a dozen of the dierential geometryarticles in the Mathematical Society of Japans excellent Encyclopedic dictionary of mathematics[34] into asmall coherent presentation in a logical order with uniform notation, together with prerequisites and furtherdetailsfromothertexts. Theoriginal targetlengthof about50pageshasunfortunatelybeenexceeded!Therecursivecatchment areaof dierential geometryprerequisites is asurprisinglylargeproportionofundergraduate mathematics. The nished product will hopefully achieve a reasonable coherence and harmonybetween the various perspectives of the subject without becoming encyclopedic.1.4.4Remark: The most dicult aspect of dierential geometry is the lack of explanation for why deni-tions are so and not otherwise. There is perhaps an analogy here with the contrast between ancient EgyptianmathematicsandclassicalGreekmathematics. It wassaidthatThalesbroughtbackmathematics fromavisit to Egypt in around 600bc. (See Ball [188], pages 1419: Probably it was as a merchant that ThalesrstwenttoEgypt, butduringhisleisuretherehestudiedastronomyandgeometry. Therehadbeenverysubstantial seatradeintheEasternMediterraneanforcenturies. Sosuchcontactswereinevitable.)The Egyptian priest class never gave reasons for why their theorems were true. They simply observed, forexample, that a 3/4/5 triangle has a right angle. (See Bell [190], page 40.)The Egyptians just said: This ishow you do it.The Greeks, by contrast, insisted on trying to nd proofs, and by nding proofs, the Greekswere able to enormously expand the body of theorems. Classical Greek mathematics was characterized bytheexcitementofdiscoverywhereasEgyptianmathematicswasstatic. (ItisjustpossiblethattheseverelimitationsoftheEgyptian writingsystemhad somethingto do withthis. TheEgyptianscribeclasshadto learn all the symbolsby rote,although theirwriting was partlyphonetically based. TheGreek writingsystemwastherstfullyphoneticsysteminhistory, whichresultedinhighgeneralliteracybecauseonlyafewsimpleprincipleswererequiredtopronounceeverywordinthelanguage. Axiomaticanddeductivethinking were an integral part of Greek culture.)Whilesomedierentialgeometrybooksdotryveryhardtomotivatethechoicesofdenitions, therearemany denitions for which it is very dicult to nd any explanation of how the choice is made. The modernmathematicians instinct is always to modify and extend denitions to see if something useful arises. In thisbook, anattemptismadetodeterminewhathappensifmanyoftheappararentlyarbitrarychoicesandrestrictions in denitions are really necessary. If it turns out that dropping a requirement or extending thedomain of an argument results in a useless or meaningless denition, this helps to clarify the meaning. Butsometimes the usual way of doing things turns out to be an obstacle in the way of further development ofthe subject. Therefore this book tries to avoid simply saying: This is how you do it.1.4.5Remark: Avegetarian