Variational Problem s in Geometr y

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 20 5

Variational Problem s in Geometr y

Seiki Nishikaw a

Translated b y Kinetsu Ab e

American Mathematica l Societ y //•? Providence , Rhod e Islan d

10.1090/mmono/205

Editoria l B o a r d

Shoshichi Kobayash i (Chair )

Masamichi Takesak i

KIKAGAKUTEKI HENBUN MONDAI by Seik i Nishikaw a

Originally publishe d i n Japanes e by Iwanam i Shoten , Publishers , Tokyo , 199 8

Translated fro m th e Japanes e b y Kinets u Ab e

2000 Mathematics Subject Classification. Primar y 53-01 , 53C21 , 53C43 , 58E20, 58J25 .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Nishikawa, Seiki . [Kikigakuteki henbu n mondai . English ] Variational problem s i n geometr y / Seik i Nishikaw a ; translate d b y Kinets u

Abe p. cm . — (Translation s o f mathematica l monographs , ISS N 0065-928 2 ;

v. 205) (Iwanami serie s i n modern mathematics ) Includes bibliographica l reference s an d index . ISBN 0-8218-1356- 0 (acid-fre e paper ) 1. Harmoni c maps . 2 . Variational inequalitie s (Mathematics) . 3 . Riemann -

ian manifolds . I . Title. II . Series. III . Series: Iwanam i serie s in modern math -ematics QA614.73.N5713 200 1 514/.74—dc21 200104635 0

© 200 2 by the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o the United State s Government . Printed i n the United State s o f America .

@ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guideline s established t o ensure permanenc e an d durability .

Information o n copying an d reprinting ca n be found i n the back o f this volume . Visit th e AMS home pag e a t URL : http://www.ams.org /

10 9 8 7 6 5 4 3 2 1 0 7 06 05 04 03 0 2

Contents

Preface t o th e Englis h Editio n vii

Preface i x

Outlines an d Objective s o f the Theor y xii i

Chapter 1 . Ar c Lengt h o f Curve s an d Geodesie s 1 1.1. Ar c lengt h an d energ y o f curves 1 1.2. Euler' s equatio n 9 1.3. Connection s an d covarian t differentiatio n 1 6 1.4. Geodesie s 2 8 1.5. Minima l lengt h propert y o f geodesies 3 8 Summary 4 3 Exercises 4 4

Chapter 2 . Firs t an d Secon d Variatio n Formula s 4 7 2.1. Th e firs t variatio n formul a 4 7 2.2. Curvatur e tenso r 5 4 2.3. Th e secon d variatio n formul a 6 5 2.4. Existenc e o f minimal geodesie s 6 9 2.5. Application s t o Riemannia n geometr y 7 7 Summary 8 2 Exercises 8 2

Chapter 3 . Energ y o f Map s an d Harmoni c Map s 8 5 3.1. Energ y o f maps 8 5 3.2. Tensio n fields 9 0 3.3. Th e firs t variatio n formul a 9 9 3.4. Harmoni c map s 10 3 3.5. Th e secon d variatio n formul a 11 0 Summary 11 4 Exercise 11 4

vi CONTENT S

Chapter 4 . Existenc e o f Harmonic Map s 11 9 4.1. Th e hea t flo w metho d 11 9 4.2. Existenc e o f loca l time-dependen t solution s 13 0 4.3. Existenc e o f globa l time-dependen t solution s 14 1 4.4. Existenc e an d uniquenes s o f harmonic map s 15 0 4.5. Application s t o Riemannia n geometr y 15 5 Summary 16 0 Exercises 16 0

Appendix A . Fundamental s o f the Theor y of Manifold s an d Functiona l Analysi s 16 3

A.l. Fundamental s o f manifolds 16 3 A.2. Fundamental s o f functiona l analysi s 17 3

Prospects fo r Contemporar y Mathematic s 18 3

Solutions t o Exercis e Problem s 18 7

Bibliography 20 1

Index 207

Preface t o th e Englis h Editio n

This book , publishe d originall y i n Japanese , i s a n outgrowt h o f lectures give n a t Tohok u Universit y an d a t th e 199 5 Summe r Grad -uate Progra m o f the Institut e fo r Mathematic s an d It s Applications , University o f Minnesota . I n thes e lectures , throug h a discussio n o n variational problem s o f th e lengt h an d energ y o f curve s an d th e en -ergy of maps, I intended t o guide the audienc e to the threshold o f th e field of geometric variational problems , tha t is , the study o f nonlinea r problems arisin g i n geometr y an d topolog y fro m th e poin t o f view of global analysis .

It i s my pleasure an d privileg e to express my deepest gratitud e t o Professor Kinets u Ab e who generously devote d considerabl e tim e an d effort t o th e translation . I woul d als o lik e t o tak e thi s opportunit y to expres s my dee p appreciatio n t o Professo r Phillip e Tondeu r wh o invited m e t o joi n th e 199 5 Summe r Graduat e Program , an d t o m y friend Andre j TVeiberg s fo r makin g hi s note s [26 ] availabl e t o th e organization o f the las t chapter .

Seiki Nishikaw a

April 200 1

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Preface

It i s sai d tha t technique s fo r surveyin g wer e develope d fro m th e need to restore land s afte r frequen t flood s o f the Nil e River i n ancien t Egypt. Geometr y i s the are a o f mathematic s whos e nam e originate s from thi s metho d o f surveying ; namely , "t o measur e lands " (ge o = lands, metr y = measure) . A s such , i t i s a n ancien t practic e t o stud y figures fro m th e vie w o f practica l applications . I t i s als o sai d tha t the ancien t Greek s alread y kne w o f the metho d o f indirec t surveyin g using th e congruenc e condition s o f triangles .

A minimal lengt h curv e joining two points i n a surface i s called a geodesic. On e may trace the origin of the problem of finding geodesies back t o th e birt h o f calculus .

Many contemporar y mathematica l problems , a s i n th e cas e o f geodesies, ma y b e formulate d a s variationa l problem s i n surface s o r in the mor e generalize d for m o f manifolds . On e ma y characteriz e th e geometric variationa l problem s a s a field of mathematics tha t studie s the globa l aspect s o f variationa l problem s relevan t i n th e geometr y and toplog y o f manifolds . Fo r example , th e proble m o f finding a surface o f minima l are a spannin g a give n fram e o f wir e originall y appeared a s a mathematica l mode l fo r soa p films. I t ha s als o bee n actively investigate d a s a geometric variationa l problem . Wit h recen t developments i n compute r graphics , totall y ne w aspect s o f the stud y on th e subjec t hav e begu n t o emerge .

This book i s intended t o be an introduction t o some of the funda -mental questions and results in geometric variational problems, study -ing th e variationa l problem s o n th e lengt h o f curve s an d th e energ y of maps .

The first tw o chapter s approac h variationa l problem s o f lengt h and energ y o f curve s i n Riemannia n manifold s wit h a n in-dept h dis -cussion o f th e existenc e an d propertie s o f geodesie s viewe d a s th e solution t o variationa l problems . I n addition , a specia l emphasi s i s

ix

x P R E F A C E

placed o n th e fac t tha t th e concept s o f connection an d covarian t dif -ferentiation ar e naturall y induce d fro m th e firs t variatio n formul a o f this variationa l problem , an d tha t th e notio n o f curvature i s obtained from th e secon d variationa l formula .

The las t tw o chapters trea t th e variational proble m on the energ y of maps between two Riemannian manifold s an d it s solutions, namel y harmonic maps . Th e concep t o f harmoni c map s include s geodesie s and minima l submanifold s a s examples . It s existenc e an d propertie s have successfull y bee n applie d t o variou s problem s i n geometr y an d topology. Thi s book takes up the existence theorem of Eells-Sampson, which is considered to be the most fundamenta l amon g existence theo-rems for harmonic maps . Th e proof use s the inverse function theore m for Banac h spaces . I t i s presented t o b e a s self-containe d a s possibl e for eas y reading .

Each chapte r o f thi s boo k ma y b e rea d independentl y wit h min -imal preparatio n fo r covarian t differentiatio n an d curvatur e o n man -ifolds. Th e firs t tw o chapters , throug h th e discussio n o f connection s and covarian t differentiation , ar e designe d t o provid e th e reade r wit h a basic knowledge of Riemannian manifolds . A s prerequisites for read -ing this book, the author assume s a few elementary fact s in the theor y of manifolds an d functiona l analysis . The y are included in the form of appendices a t th e end o f the book . Detail s i n functional analysi s ma y be skipped . Th e reader , however , i s encourage d t o d o th e exercis e problems a t th e en d o f eac h chapte r b y himsel f o r hersel f first . Th e solutions ma y b e consulte d i f necessary , sinc e man y o f th e exercis e problems complemen t th e content s o f the book .

This book is an outgrowth of lectures delivered a t Tohoku Univer -sity an d th e 199 5 Summer Graduat e Program s hel d a t Th e Institut e for Mathematic s an d It s Applications , Universit y o f Minnesota . Th e first hal f o f th e boo k aim s a t a junior an d senio r level , an d th e las t half a t a firs t an d secon d yea r graduat e level . Eac h hal f roughl y con -sists o f th e amoun t o f topic s tha t ma y b e covere d i n on e semester . In th e actua l lectures , th e autho r als o discusse s th e harmoni c map s between Rieman n surfaces . Thi s portio n i s not include d i n this boo k due to the limite d space . Th e reade r wh o is interested i n the study of harmonic map s i s advise d t o firs t stud y th e harmoni c map s betwee n Riemann surfaces .

It woul d b e thi s author' s wis h a s wel l a s pleasur e i f thi s boo k could interes t man y reader s i n variationa l problem s i n geometry .

PREFACE x i

Last bu t no t least , th e autho r expresse s hi s sincer e gratitud e t o the editoria l staf f o f Iwanam i Sho t en fo r thei r valuabl e hel p i n th e publication o f thi s book .

Seiki Nishikaw a

December 199 7

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Outlines an d Objective s o f th e Theor y

Among geometri c variationa l problems , th e extrem e valu e prob -lem regarding the length of curves is as old as those in calculus. Chap -ter 1 of this boo k i s devoted t o discussion s o f variational problem s o f curves i n manifolds . A s i s wel l known , th e lengt h o f a curv e joinin g two points in a plane is given by integrating the magnitude o f tangen t vectors. Similarly , on e can defin e th e lengt h an d energ y fo r curve s i n more genera l Riemannia n manifold s b y measurin g th e magnitud e o f the tangen t vector s usin g Riemannia n metrics .

In Chapte r 1 , Euler' s equatio n i s calculated . I t characterize s the critica l point s o f th e lengt h an d energ y o f curve s whe n the y ar e considered as functionals define d i n the space of curves. Consequently , the equatio n o f geodesie s i s obtained . Th e concept s o f connection s and covarian t differentiatio n ar e naturall y induce d fro m th e equatio n of geodesies in a manifold. Covarian t differentiation , a n essentia l too l for studyin g variationa l problem s i n manifolds , i s a n operatio n tha t defines th e derivativ e o f a vector field by a vector field in a manifold .

The mos t fundamenta l connection , calle d th e Levi-Civit a con -nection, i s uniquel y determine d i n a manifol d equippe d wit h a Rie -mannian metric , i.e. , a Riemannia n manifold . Th e notio n o f paralle l transport i s induced from thi s connection. Th e discovery of the notio n of paralle l transpor t i n Riemannia n manifold s (1917 ) an d Einstein' s use o f geometry base d o n a four-dimensiona l indefinit e metri c fo r hi s general relativit y (1916 ) greatl y advance d th e stud y o f Riemannia n geometry.

Geodesies i n Riemannia n manifold s correspon d t o straigh t line s in th e plan e an d the y ar e locall y characterize d a s th e curve s o f min -imal lengt h betwee n points . On e ca n construc t a specia l loca l coor -dinate system , calle d a norma l coordinat e system , usin g thes e mini -mal geodesie s abou t eac h poin t i n a Riemannia n manifold . Paralle l transport an d norma l coordinat e system s ar e th e mos t basi c tool s i n

xi i i

xiv OUTLINE S AN D O B J E C T I V E S O F T H E THEOR Y

comparing the geometry of a Riemannian manifol d wit h the geometr y of a mode l spac e (fo r example , Euclidea n space) .

In Chapte r 2 , usin g covarian t differentiation , th e first variatio n formula (Euler' s equation ) fo r th e variationa l proble m regardin g th e energy o f curve s i n Riemannia n manifold s i s compute d i n th e gen -eral cas e wher e th e imag e o f a curv e i s no t alway s containe d i n a local coordinat e neighborhood . Th e secon d variatio n formul a i s sub-sequently computed . Jus t a s the notion of connections i s derived fro m the first variatio n formula , i t i s seen that th e second variation formul a possesses a n intimat e relationshi p t o th e notio n o f curvatur e i n Rie -mannian manifolds . I n othe r words , th e notion s o f curvatur e tenso r and th e curvatur e o f a Riemannia n manifol d ar e naturall y induce d from th e secon d variatio n formul a fo r th e energ y o f curves .

Given two points in a Riemannian manifold , th e distance betwee n these tw o point s i s give n b y th e leas t uppe r boun d o f th e length s o f piecewise smoot h curve s connectin g them . Whethe r a Riemannia n manifold become s a complet e metri c spac e wit h respec t t o thi s dis -tance i s an importan t question . I t wa s relatively recentl y (1931 ) tha t Hopf-Rinow gav e necessary an d sufficien t condition s fo r th e question . The result s b y Hopf an d Rino w are significant no t onl y in making th e notion o f completenes s succinct , bu t als o i n showin g tha t thi s com -pleteness i s the conditio n tha t guarantee s th e existenc e o f a minima l geodesic joining tw o give n points .

As state d above , th e secon d variatio n formul a fo r th e energ y o f curves i s closel y relate d t o th e curvatur e o f Riemannia n manifolds . Using this , on e ca n stud y th e effect s o f th e curvatur e o f a Riemann -ian manifold o n its topological structure. Myers ' theorem an d Synge' s theorem ar e discusse d a s typica l example s o f suc h applications . Th e former state s tha t th e fundamenta l grou p o f a compac t an d con -nected, Riemannia n manifol d o f positiv e curvatur e i s a finit e group , and th e latte r state s tha t a n even-dimensiona l compact , connecte d and orientabl e Riemannia n manifol d o f positiv e curvatur e i s simpl y connected. Researc h o n Riemannia n manifold s usin g existenc e an d properties o f geodesie s i s being activel y pursued .

In Chapte r 3 , harmoni c map s an d th e energ y o f map s ar e dis -cussed. The y generaliz e th e variationa l proble m o f th e energ y o f curves i n Riemannia n manifolds . Namely , a functiona l calle d th e energy o f maps i s defined i n th e mappin g spac e consistin g o f smoot h maps between Riemannian manifolds , an d harmonic maps given as its

OUTLINES AND OBJECTIVE S O F TH E THEOR Y x v

critical point s ar e investigated . Th e energ y o f map s i s a natura l gen -eralization o f th e energ y o f curves . Example s o f harmoni c map s ap -pear i n variou s aspect s o f differentia l geometry . Harmoni c functions , geodesies, minima l submanifolds , isometri c maps , an d holomorphi c maps ar e a fe w typica l examples .

The first variatio n formula , whic h characterize s th e critica l point s of th e energ y functional , ca n b e obtaine d b y essentiall y th e sam e approach a s i n th e cas e o f geodesies . However , th e computation s become unnecessaril y complicate d an d onl y yiel d result s o f a loca l nature withou t us e o f th e covarian t differentiatio n tha t i s naturall y induced fro m th e Levi-Civit a connectio n o f Riemannia n manifolds . To alleviat e thes e difficulties , i t i s designe d i n thi s chapte r t o derive , through discoverie s i n th e process , th e computationa l rule s fo r th e covariant differentiatio n tha t i s induce d fro m th e Levi-Civit a connec -tion i n tangen t bundle s an d thei r tenso r product s ove r Riemannia n manifolds. Thi s rout e ma y no t b e th e mos t direc t one , bu t th e autho r believes tha t i t i s mor e effectiv e i n familiarizin g th e reade r wit h th e definition an d th e rule s o f computation s fo r covarian t differentiatio n than th e axiomati c approach . A t first, th e reade r ma y fee l uneasy , especially abou t th e portio n o f th e induce d connections . Nonethe -less, actua l computation s hel p promot e understandin g o f th e notion . The fastes t wa y t o gras p th e rule s o f computatio n involvin g covarian t differentiation i s actuall y t o engag e i n th e computations . Th e com -putations o f th e first variatio n formul a fo r th e energ y functiona l o f maps yiel d a vecto r field calle d th e tensio n field. I t i s give n a s th e trace o f th e secon d fundamenta l for m o f th e maps . A harmoni c ma p is the n characterize d a s a ma p whos e tensio n field i s identicall y 0 .

Chapter 4 i s devote d t o th e existenc e proble m o f harmoni c map s between compac t Riemannia n manifolds . Whethe r o r no t a give n map i s homotopicall y deformabl e t o a harmoni c ma p i s on e o f th e most fundamenta l question s amon g geometri c variationa l problems . It ma y b e regarde d a s a generalizatio n o f th e existenc e proble m o f closed geodesies . T o thi s end , th e "hea t flow method " i s first intro -duced. Thi s i s a n effectiv e techniqu e fo r deformin g a give n ma p t o a harmonic map . Then , usin g thi s technique , i t i s prove d tha t an y ma p from a compac t Riemannia n manifol d M int o a compac t Riemannia n manifold T V o f nonpositiv e curvatur e i s free homotopicall y deformabl e to a harmoni c map . Thi s theore m wa s first prove d b y Eells-Sampso n in 1964 .

xvi OUTLINE S AN D O B J E C T I V E S O F T H E THEOR Y

The proo f o f thi s theore m usin g th e hea t flow metho d firs t re -quires th e existenc e o f a time-dependen t solutio n t o a n initia l valu e problem wit h an y initia l ma p o f th e paraboli c equatio n fo r harmoni c maps. Th e origina l proo f use s successiv e approximation s t o construc t a solutio n afte r convertin g th e proble m t o a proble m o f integra l equa -tions vi a th e fundamenta l solutio n o f th e hea t equation . I n thi s book , the solutio n i s constructed throug h us e o f the invers e functio n theore m in Banac h space s i n a n effor t t o minimiz e th e amoun t o f preparation .

The existenc e o f time-dependen t loca l solution s i s alway s guar -anteed, bu t th e existenc e o f globa l time-dependen t solution s i s no t self-evident, sinc e th e paraboli c equatio n fo r harmoni c map s i s nonlin -ear. I n fact , provin g th e existenc e o f globa l t ime-dependen t solution s entails som e estimate s o f th e growt h rat e o f solution s i n time . Th e curvature o f th e Riemannia n manifol d N play s a crucia l rol e i n esti -mating th e influenc e o f nonlinea r terms . A n estimatio n formul a tha t guarantees th e existenc e an d convergenc e o f time-dependen t globa l solutions i s obtaine d usin g th e Weitzenboc k formul a fo r th e hea t op -erator unde r th e conditio n tha t N i s o f nonpositiv e curvature .

The Weizenboc k formula , i n general , give s th e relationshi p be -tween secon d orde r partia l differentia l operator s naturall y actin g o n tensor fields o n Riemannia n manifold s an d th e Laplac e o r hea t oper -ator actin g o n functions . I t i s reveale d tha t th e Rieman n curvatur e and it s Ricc i identit y pla y essentia l role s fo r existenc e o f solution s t o those differentia l operators . I n thi s chapter , a n a prior i estimat e re -garding th e growt h rat e o f solution s i s obtained usin g th e Weizenboc k formula fo r th e energ y densit y o f solution s t o th e paraboli c equatio n for harmoni c map s an d th e hea t operator . Thi s ide a i s originall y du e to Bochner . I t ha s becom e a n effectiv e an d fundamenta l techniqu e for th e proof s o f theorems suc h a s th e Kodair a vanishin g theore m an d more recentl y i n gaug e theory .

As i n th e cas e o f geodesies , on e ca n als o investigat e th e struc -tures o f Riemannia n manifold s usin g th e existenc e an d propertie s o f harmonic maps . Th e theore m o f Preissman , on e o f th e typica l ap -plications o f harmoni c maps , i s discussed . Th e theore m state s tha t a nontrivia l Abelia n subgrou p o f th e fundamenta l grou p o f a com -pact manifol d o f negativ e curvatur e i s infinitel y cyclic . Th e researc h of Riemannia n manifold s usin g th e existenc e an d propertie s o f har -monic map s seem s t o posses s a promisin g future . Fo r example , ne w proofs fro m a mor e analytica l poin t o f vie w fo r th e topologica l spher e theorem an d th e Franke l conjectur e wer e recentl y give n b y exploitin g

OUTLINES AN D OBJECTIVE S O F TH E THEOR Y x v i 1

the existenc e theore m o f harmoni c sphere s du e t o Sack s an d Uh -lenbeck. A stron g rigidit y theore m regardin g comple x structure s i n Kahler manifold s o f negativ e curvatur e wa s als o obtaine d usin g th e existence theore m o f Eell s an d Sampson .

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Solutions t o Exercis e Problem s

Chapter 1

1.1 Easil y follow s fro m th e definition . 1.2 Le t {Vp} b e an open cove r of M suc h that eac h Vp is a loca l

coordinate neighborhoo d o f M an d Vp i s compact . Sinc e M wit h the secon d axio m o f coun t ability i s paracompact , ther e ar e a locall y finite refinemen t {U a} an d a partitio n o f unit y {f a} subordinat e t o {Ua}. Sinc e U a ca n b e identified wit h a n ope n subse t o f Rm throug h the loca l coordinat e system , i t ha s a Riemannia n metri c g a. A t eac h point x G M, defin e a n inne r produc t g x i n T XM b y

gx(v,w) = ^2 fa(x)g^(v,w), v,w e T XM.

Here, J ^ represent s th e su m ove r a wit h x G U a. The n g = {g x} is th e desire d Riemannia n metric . Ther e i s anothe r wa y t o prov e it. Not e the theore m o f Whitney tha t state s tha t an y m-dimensiona l C°° manifol d ca n be imbedded in 2ra-fl-dimensional Euclidea n space. Then follo w Exampl e 1.2 .

1.3 D o i n th e sam e manne r a s derivin g (1.14 ) fro m (1.13) . 1.4 (1 ) Th e transformatio n formul a o f the natura l fram e fields

d ^ dx q d d ^ dx r d dxi ^ dx~i dx q' dx k ^ dx k dx r

q—l r—1

and (1.23 ) yiel d

A dx k ^ , \ , d *J V dx*dx k + ^ - f qr dx k I [ dx p' p=l ^ <?=1 \ q=l / )

Substitute the following transformation formul a i n the above equatio n

d ^ dx % d dxP ~ ^ dxP dx l

i=l

and compar e i t wit h (1.22) .

187

188 SOLUTIONS T O EXERCIS E PROBLEM S

(2) Defin e V x F b y (1.23 ) i n each coordinat e neighborhood . The n we ca n patch the m togethe r usin g th e transformation formula s i n (1); consequently, a linea r connectio n V i s obtaine d i n M .

1.5 (1 ) For the second equation , w e replace AT , Y, Z i n [A , [Y, Z]] = [A , YZ - ZY] = XYZ - XZY - YZX + ZYX i n th e orde r X -+ Y - > Z - > A .

(2) Fro m th e firs t equatio n i n (1) , we se e tha t th e Li e derivativ e LxY satisfie s th e followin g rules :

(i)Lx(Y + Z) = L xY + L xZ, (ii) L x(fY)=X(f)Y + fL xY, (ui) L X+YZ = L XZ + L YZ, (iv) L fxY = fL xY - (Yf)X.

Among them , (iv ) i s differen t fro m th e rul e VfxY = fVxY fo r covariant differentiation . Thi s implie s tha t th e valu e LxY(x) o f th e Lie derivativ e i s essentially determine d b y the behavior o f both X an d Y aroun d x, unlik e th e covarian t derivative , wher e th e valu e VxY i s determined b y th e behavio r o f Y aroun d x an d th e valu e X(x) o f X at x. I n othe r words , Li e differentiation i s not a n operato r determine d by th e directio n X(x) o f differentiatio n alone .

Furthermore, th e second equatio n i n (1 ) yields a relation L xLy — LyLx = L[X,Y]I bu t th e relatio n V j V r — V y Vx = V[X,Y] doe s no t generally hol d fo r covarian t differentiation . Thi s las t relatio n hold s only whe n th e Riemannian manifold s ar e flat (curvatur e tenso r R = 0 ) (see §2.1) .

1.6 Le t { e i , . . . , em } b e a bas e fo r T XM. Se t Ei(t) = Ptei ( 1 < i < m). Sinc e {Ei(t), ... , Ei(t)} i s a bas e fo r T C^M fo r eac h £ , we can writ e a s

m

2 = 1

Then, fro m th e propertie s o f linear connection s an d E x bein g parallel , we ge t

m dY % d ( m

1.7 I f w e assum e tha t ther e i s a vecto r field $ a s state d i n the problem , w e se e tha t it s integra l curve s <p(t) = (c(t) ,c r(t)) ar e solutions t o (1.30 ) and , hence , unique . A s fo r it s existence , w e ma y locally defin e th e vecto r fiel d $ b y (1.30) . Fro m th e uniqueness , $ i s defined globally .

/—n ltPflY^

y:=n

SOLUTIONS T O EXERCIS E PROBLEM S 189

1.8 (1 ) Necessity i s clear fro m th e definition . Sufficienc y follow s from th e uniquenes s o f geodesie s regardin g th e initia l condition . (2 ) follows readil y fro m (1) .

1.9 A s i s wel l known , a connecte d C°° manifol d i s arc-wis e connected. Hence , ther e i s a continuou s curv e c : [a , b] —* M joinin g p an d q. Cove r th e compac t se t cda^}) C M b y a finit e numbe r o f coordinate neighborhood s U a, the n replac e c by a C°° curv e i n eac h Ua.

1.10 Give n p G M , fo r sufficientl y smal l e , exp p JBe(0) = {q e M | d(p,q) < e} holds , wher e B e(0) = {v e T p M | |v | < e} . expp i s a loca l homeomorphis m an d exp p Z? e(0)'s for m a bas e fo r th e local neighborhoo d syste m a t p. Consequently , th e topolog y define d by d coincide s wit h th e topolog y o f the manifold .

C h a p t e r 2

2.1 Thi s follow s readil y fro m th e definition . 2.2 A typ e (1 , s) tenso r field

r = X ) ^ i . - - i f l (d a ; i l ) ® ' - - ® ( d a ; i a ) a

on T pM correspond s t o a n s linea r ma p

( - ) \dxJ

°-\ (J-W^^.-f^ dx^Jx

1'" ' V&rVJ V l1 '"1* \dxJ J

from T X M x • • • x T X M int o T XM. 2.3 Thi s follow s because , i n general , V(XJY) = (Xf)Y +

fVxY^fV(X,Y). 2.4 Sinc e [d/dx l, d/dx^] — 0, fro m th e definition , w e hav e

A J M J L - v V — - V V — 9x*' <9x J y ftr fc 9 ^ ^J dx k a^ i a ^ &E fc

171 ( f) r) 7Jl ^ f)

E J _?_w 5_ w + V < r ' rr r z r r u I dx* - ^ Q xj

lk Z-^ K ir i k o r lk) \ dx r

1=1 ^ r=l )

2.5 I t ca n b e don e readil y b y notin g th e propertie s o f R i n Proposition 2.1 1 and that th e orthonormal basi s {v f, w'} fo r a i s given by th e Gram-Schmid t orthonormalizatio n a s

„,/ _ v , __ w - g x(v,w)v'

\v\ \w - g x(v,w)v'[

190 SOLUTIONS T O EXERCIS E PROBLEM S

2.6 I t suffice s t o b e abl e t o determin e R(u, v, w, £) , Vix , v,w,t G TXM, give n R(u,v,v,u), Vu,v G TXM. Firs t o f all , fro m th e relatio n

R(u + t,v,v,u + t) = R(u, v, v, u) + R(t, v, v, t) + 2R(u, v, v, t)

follows tha t R(u,v,v,t) i s also determined . Fro m thi s an d th e relatio n

R(u, v + w, v + w, t) = R(u, v, v, t) + R(u, w, w, t)

+ R(u, v, w, t) + R(u, w, v, t ) ,

we hav e R{u, v, w, t) + R(u, w, v, t) = (*) ,

where (* ) i s th e su m o f know n terms . B y applyin g th e first Bianch i identity t o th e secon d term , w e ge t

2R(u, v, w, t) - R(w, v, u, t) = (*) .

By exchangin g u an d w here , w e als o ge t

2R(w, v, u, i) — R(u, v, w, t) = (*) .

These tw o equation s impl y tha t R(u, v, w, t) — (*); hence , R(u, v, w, t) is determined .

2 .7 Conside r a C°° ma p u : O - • M wit h u(0,s) = u(0,0 ) (—e < s < e) , wher e O i s a n ope n subse t o f M 2 give n b y

O = {(£ , s)\ - e < t < 1 + e , - e < s < e} ( e > 0) .

Given v G T XM, defin e a C° ° vecto r field V alon g u t o satisf y tha t V(0, s) = v an d V(£ , 5) i s th e paralle l t ranspor t o f v alon g eac h C°° curve £ 1—» ii(t , s ) fo r t 7 ^ 0. The n b y Lemm a 2.15 , w e ge t

ds dt dt ds \ds dt J

Since th e paralle l t ranspor t doe s no t depen d o n th e choic e o f th e curves fro m th e assumption , V(l,s) equal s th e paralle l t ranspor t o f

1/(1,0) alon g th e C° ° curv e s H- » u(l,s); hence , — F(1 ,0 ) = 0 . Con -a s

sequently, fro m th e abov e equation , w e ge t

fl(^(l,0),^(l,0))v(l,0)=0.

Since u an d v ar e arbitraril y chosen , w e ge t th e desire d conclusion . 2.8 Se t 'V XY = d^' 1 {V d ^{x)dLp(Y)). The n ' V define s a lin -

ear connectio n i n M an d satisfie s condition s (i ) an d (ii ) i n Theore m 1.12. Fro m th e uniquenes s o f th e Levi-Civit a connections , w e se e tha t 'V = V . Thi s implie s (1) . (2 ) readil y follow s fro m (1) .

SOLUTIONS T O EXERCIS E PROBLEM S 191

2.9 Fo r (1) , defin e g = Co*g. (2 ) readil y follow s fro m (2 ) o f Problem 2. 8 above .

2.10 I f M i s orien t able, Theore m 2.2 6 implie s tha t M i s sim -ply connected . I f M i s no t orientable , appl y Theore m 2.2 6 t o th e orient able doubl e coverin g spac e M.

Chapter 3

3.1 Sinc e th e suppor t o f / i s compact , th e righ t han d sid e i s a finite sum . Le t {Vp, i/jpjp^B b e anothe r coordinat e neighborhoo d system an d denot e b y {ap}^ eB a partitio n o f unity . The n a t p G [/ anV^, w e have

V/det(5fi)(p) = | det J(4> a o ^ 1 ) ( ^ ( p ) ) | y / d ^ g ) ( p ) .

The chang e o f variabl e formul a fo r th e integra l i n R m the n readil y implies

E / Upfy/detigl,)) o ^dx\ • • • dxj

= E / (vpPafy/detigZi)) o ^xdx\ • • • dxj

= E / (<rpP*fyfet>(9ki)) a^J^(v0nua)

o if;'1] det J(<t> a o ^ 1 ) ! ^ • • -dx™

= E / U/3PafJdet(g^)) o <^dx\ ->-dx r0

ajpJ<t>a(u0nua)

= E / (PafJdet(g^)) o ^dxi • • • dx1?.

That jji g is positive definite , namely , n g(f) > 0 for nonnegativ e func -tions / , i s clear fro m th e definition . I n orde r t o se e \i g bein g a Rado n measure o n M , namely , bein g a bounded linea r functiona l i n Co(M) , we mus t sho w th e following . Give n a n arbitrar y continuou s functio n / whos e suppor t i s a compac t subse t K, ther e i s a constan t CK such that

M / ) | < c K su p 1/0) 1

holds. Thi s i s also clea r fro m th e definition .

192 SOLUTIONS T O EXERCIS E PROBLEM S

3.2 Choos e a coordinat e neighborhoo d U o f x £ M an d a coordi -nate neighborhoo d V o f u(x) £ N s o tha t u(C7 ) C V holds . Denot e b y (ya) a loca l coordinat e syste m i n V. Sinc e {(d/dy a) ou} (1 < a < n) forms a basi s fo r th e fiber T U^N ove r x i n u~ 1TN1 a sectio n rj o f u~1TN ove r L 7 is expresse d a s 7 7 = ^ a r) a(d/dya) ou. Assum e tha t a linear connectio n 7 V i n th e proble m exists . Then , fro m th e propertie s of a connection , w e mus t hav e

and, hence , i s unique . As fo r th e existence , w e ma y defin e ; V b y th e abov e equation .

It i s readil y verifie d tha t 'V , independen t o f th e choic e o f coordinat e systems, define s a linea r connection .

3.3 Eac h ca n b e verifie d directl y fro m th e definitions . Fo r (2) , see th e solutio n t o Proble m 3. 4 below .

3.4 Le t {(ft} b e th e one-paramete r grou p o f transformation s generated b y X. Denot e b y (x l) a loca l coordinat e syste m i n th e coordinate neighborhoo d (U,(j)). Th e measur e determine d b y th e pull -back (f^g o f g unde r cp t is give n a s

d/Vt*<

N A<*-\Y.^'"f'^V,~j ]dx 1---dx'

Hence, w e hav e

— I dt \t=o

dfi^g = d i v X • dji g.

In fact , whe n (a>ij)(t) i s a differentiabl e nonsingula r matri x wit h re -spect t o t , th e derivativ e o f th e determinan t i s give n b y

de t (a i j ( t ) ) = d e t ( a i j ( t ) ) ^ a f c l ( t ) — a w ( t ) , dt

k,l

where (a ZJ(£)) denote s th e invers e matr i x o f (a^-(t)) . Furthermore ,

since d/dt\ t=o (d^/dx 1) = dX k/dxz an d dip^/dx 1 = 5 k fo r

SOLUTIONS T O EXERCIS E PROBLEM S 193

X — J2i Xld/dxz, w e hav e

H ^ / d e T ^ " ) V W'dt det ( ^P- \ \ dx l • • • dx m

t=o \ dx

• • • dx

= fe V*X j ^det(g lJ)dx1-"dxm

= R.H.S .

Since each ip t is a diffeomorphism, w e see readily fro m definitio n tha t

M 7 M

Consequently, b y addin g th e loca l forms o f the abov e resul t usin g th e partition o f unity , w e ge t

0 = — / dji rg = / dWXd/Lig. dt\t=oJM * ty J M

3.5 (1 ) follow s readil y fro m th e definition . T o se e tha t V T becomes a n (r , s + 1 ) tensor , it , fo r instance , suffice s t o not e tha t V T becomes C°°-lmea r wit h respec t t o C°° module s r (TM) , r(TM*) ; namely, th e followin g hold s

VT{fX,f1Xl,... ,f 3X3,hlWl,... ,h rwr)

= f fi • • • fshi • • • hrVT(X, X lt... ,X s,wi,... ,w r).

(2) follow s fro m a simila r computatio n t o Lemm a 3.4 , notin g (3.16), (3.26) .

3.6 Fro m Proble m 3. 5 above , Vi ? i s defined b y

VR(X, Y, Z, V, w) = X • R(Y, Z, V, w) - R{V XY, Z, V, w)

- R(Y, V XZ, V, w) - R(Y, Z, VXV, w)

-R(Y,Z,V,Vxw).

Noting tha t R(Y, Z, V, w) = w(R{Y, Z)V) an d V* xw{Y) = Xw(Y) -w(VxF) , w e ge t

VR(X, Y, Z, V, w) = w(Vx • R(Y, Z)V - R(V XY, Z)V

- R{Y, V XZ)V - R{Y, Z)W XV).

194 SOLUTIONS T O EXERCIS E PROBLEM S

Hence, i t suffice s t o se e tha t th e symmetri c su m ove r X, Y, Z o f th e expression

Vx • R(Y, Z)V - R{V XY, Z)V - R{Y, V XZ)V - R(Y, Z)V x V

= [V X,R(Y, Z)}V - R(V XY, Z)V - R(Y, V XZ)V

equals 0 . Here , notin g tha t V X Y - VyX = [X,Y] an d R(X,Y)Z = [Vx,Vy] - V[ X,y], w e ge t

[VX,R(Y, Z)]V - R(V XY, Z)V - R(Y, V XZ)V

+ [Vy , R{Z, X))V - R{V YZ, X)V - R{Z, V YX)V

+ [V z, R{X, Y))V - R(V ZX, Y)V - R{X, V ' Z Y)V

= [V XjR(Y, Z)]V + [Vy , R(Z, X)]V + [Vz , R(X, Y))V

- R([X, y ] , Z)V - R([Y, Z},X)V - R([Z, X},Y)V

= ( [V x, [Vy , Vz]] + [Vy , [Vz, Vx]] + [Vz , [Vx, Vy]]) F

+ (V[[x,F],Z ] + V[[y 5Z],X] + V[[z,X],F])^ -

Hence, we get th e desire d conclusio n fro m th e Jacob i identit y fo r th e operator an d th e vecto r fields .

3.7 (1) , (2 ) follo w readil y fro m th e definitions . (3) Appl y th e definitio n (1.26 ) o f th e Levi-Civit a connectio n to -

gether wit h (1) , (2) . 3.8 On e ma y verif y tha t (f) satisfie s th e equatio n fo r harmoni c

maps by a direct computation . On e can also show that 0 is a harmonic map fro m Propositio n 3.1 7 b y notin g tha t S 3 —> • S 2 i s a Riemannia n submersion an d tha t <p^ 1{y) is a geodesi c ( a grea t circle ) o f S 3 fo r each y G S 2.

3.9 Sinc e ip* = e 2pg,

^ dip k dip 1 2

k,l

and

E*<*>££g-«-VM

SOLUTIONS T O EXERCIS E PROBLEM S 195

hold fo r th e component s o f g with respec t t o a loca l coordinat e system . From thi s w e ge t

e(uo<p) = Y^ ^29 t3ha,f3{uoip)

ij,k,l a,/ 3

dua dcp k du$ dtp 1

dxk dx l dx l dxi

-2"EE^MVM-^§S^ = ̂ 2M«)M, k,l a,P

dxk dx l

<P*(dfJ>g k,l

dx1dx2 = e 2pdjig.

\(«72) From thi s i t follow s tha t follow s ((w)^ /Z) < C 3e^-a'^2\w\{^a/2)

through a simple computatio n fo r I , 7 G Y(TM). Consequently , fro m the definitio n o f tensio n field an d Lemm a 3.3 , w e ge t tha t r(cp) = (2 — m) g radp , wher e m i s th e dimensio n o f M . I n particular , whe n m = 2 , (f i s a harmoni c map .

3.10 Le t h = u*h denot e th e induce d metri c fro m h b y u an d denote b y h^ th e component s o f h. Le t fij denot e th e component s of th e typ e (1,1 ) tenso r field correspondin g t o th e typ e (0,2 ) tenso r field h o f M unde r th e ispomorphis m TM* (g ) TM*. The n w e ge t h) ~ Sf c hjk9 kt- O n th e othe r hand , w e se e tha t

< - trac e (h)) 2 J

holds fo r th e (2,2 ) matri x (fij), an d tha t th e equalit y hold s whe n

there i s a positiv e numbe r A such tha t h %- = A^ ; namely , onl y whe n

fij = Xgij. Exercis e 3.1 0 follow s readil y fro m this . On th e othe r hand , notin g tha t th e induce d connectio n ' V o n th e

vector bundl e (p~ 1TM i s compatible wit h th e fiber metri c <p* togethe r with ip*g = e 2pg an d (1.26) , w e ca n readil y verif y tha t

'Vxd<f(Y) = MVxY) + (Xp)Y + (Yp)X - g(X, Y) g r a d p

through a simpl e computatio n fo r X , Y G T(TM). Consequently , w e get tha t r((f) = ( 2 — m) g radp , wher e m i s th e dimensio n o f M. I n particular, whe n m = 2 , cp i s a harmoni c map .

196 SOLUTIONS T O EXERCIS E PROBLEM S

Chap te r 4

4.1 (1 ) Give n T eT(TM* ftu^TN) an d X , Y, Z G r (TM) , w e have

(VVT)(X, Y, Z) - (V xVT)(Y, Z )

= V X(VT(Y, Z) ) - VT(V XY, Z) - VT(Y, V XZ)

= Vx((VyT)(Z) ) - (V V x yT)(Z) - (V YT)(VXZ)

= ( V x ( V y T ) ) ( Z ) - ( V V x y r ) ( Z ) .

(2) Fro m th e definitio n o f th e connectio n i n TM* eg ) u~ 1TN, w e have

'Vy( r (Z) ) = (Vy)(Z ) + T(VyZ) ,

'Vx 'Vy(T(Z)) =(V X VyT)(Z) + (VyT)(V XZ),

+ (V XT)(VyZ) + (V XT)(VyZ).

Similarly, computin g - 'V y V; r (T(Z) ) an d - 'V [ x ,y](T(Z)) an d adding the m together , w e ge t

RV{X, Y)(T(Z)) = (R V(X, Y)T)(Z) + T{RM(X, Y)Z).

(3) Fro m (2 ) an d th e definitio n o f the induce d connectio n 'V , w e get

dxj J ) \dx k \ d x l ' dxJ ) ^ ) \dx

\dxv dxi) \ \dx k)) \ ydxi'dxi) dx k)

a \/3, 7,<5 Z / y

On th e othe r hand , sinc e w e hav e

V V T ( ^ ^ ' ^ ) = ? v ^ T f e Q ^ o u ' the conclusio n follow s fro m th e definitio n o f R v.

4.2 W e may us e an ide a similar th e on e used i n the proo f o f th e Weitzenbock formul a fo r e{u t). First , fro m th e definitio n o f ft(ut), w e have

dn{ut) ^n,a ^ '

a,(3

SOLUTIONS T O EXERCIS E PROBLEM S 197

dua

On the other hand , notin g tha t V / - ^ - = V^Vjii" , w e get

k,l a,(3

V — dt

|2

From the Ricci identity regarding the connection V in T(Mx(0, T))*(g) u~1TN, w e then get

VifeVtVz<-VtVf cVf c<

^ ktl dxr ^ ^ 7(5 e axfc a t cb z' Noting tha t M x (0, T) is a product o f Riemannian manifolds , w e can readily verif y tha t R Mx(0,T\tl = 0 from th e definition o f the curva -ture tensor . Consequently , w e get the desired resul t b y substitutin g the Ricc i identit y i n th e abov e equatio n an d b y notin g tha t u i s a solution t o the equation fo r harmonic maps .

4.3 I f we consider the derivative dexp^ p0^ o f the map exp : U —> N a t (p , 0) G TML followin g th e line alon g th e proof s o f Theorem s 1.24 an d 1.25 , we see that it s matri x representatio n wit h respec t t o the canonica l coordinat e syste m give s rise to

10 01

Hence, noting that M i s compact, the existence of the desired e follows from th e inverse functio n theorem .

4.4 Notin g Lemm a 3. 3 and the definitio n o f the induce d con -nection, w e get, for X,Y e T(TMi) ,

Vd(/2 o A)(X, Y) = Vx(d/ 2 o dh{Y)) - d(f 2 o dh)(V xY)

= ^d fl(x)df2)(dMY)) + df2(Vx(dfi(Y))) - df 2odf1(VxY)

= Vdf 2(df1(X),df1(Y)) + df 2(Vdf1(X,Y)).

From thi s follow s th e first equation . Th e secon d equatio n readil y follows fro m th e first equation .

4.5 I f S/du — 0, the formul a fo r th e secon d fundamenta l for m of composition map s implie s tha t

^ d(uoc) 7 / dc\ ^ 7 f dc dc\ Vd^Aw2=du ( v ^ ) + V d u U ' jt) = °

198 SOLUTIONS T O EXERCIS E PROBLEM S

for an y geodesi c c : / — • M. Fro m thi s follow s (i ) = > (ii) . Conversely , if (ii ) holds , S/du(dc/dt, dc/dt) = 0 hold s fo r th e tangen t vecto r dc/dt of th e geodesic . Vdu = 0 holds , sinc e an y tangen t vecto r v G TM a t each poin t x G M ca n b e give n i n th e for m o f dc/dt fo r som e geodesic .

4 .6 Se t Q = M x [ 0 , T ] . Fo r example , regardin g C 2 + a ' 1 + a / 2 ( Q , R ^ ) being a Banac h space , a proo f goe s a s follows . Le t {uk} b e a Cauch y sequence i n C 2 + c * ' 1 + a / / 2 ( (3 , l ^ ) - Sinc e {uk} i s a uniforml y bounde d and equicontinuou s sequenc e i n C 2 , 1 ( (3 ,R g ) , th e Ascoli-Arzel a theo -rem implie s tha t ther e i s a subsequenc e {uk>} o f {uk} suc h tha t i t converges t o som e u i n C7 2,1(Q,lRg). The n i t suffice s t o sho w tha t u is a n elemen t o f C 2 + a ' 1 + a / 2 ( Q , R 9 ) an d tha t {uk} converge s t o u i n C2+a>1+a/2(Q^M.q). Thes e ca n b e directl y verifie d fro m th e definitio n

r , i I i ( 2 + a , l + a / 2 ) of th e nor m \U\Q .

4 .7 Sinc e \Cw\%''"' /2) = I C H Q + ( C ^ ) ^ + {(w) xa'/2) fro m

the definitio n o f th e norm , i t suffice s t o estimat e individuall y | C H Q >

((w)x an d ((w)x • From th e assumption , w G C 2 , 1((2,M9) an d

w(x1 0 ) = 0 , w e se e tha t | C H Q < C i e a / 2 | C u ' | J , a / 2 ) . Similarly , w e se e

tha t ((w)i a,) < C 2e(a-a'^2\Cw\(Q'a/2). (Fo r example , w e ma y trea t

the tw o case s wher e d(x,x') > e 1/2 an d d(x,x') < e 1/2.)

On th e othe r hand , regardin g (£w)x , fo r 0 < t < t' < 2e , w e have

\at)w(x,t)-at')w(x,t')\ < \at)w(x,t) - w(x,t')\ + |(C(t ) - C(t'))(w(x,t) - w(x,0))\

< ICWIHQ' Q / 2 ) l* - *T /2 + 2e_1|t - t ' |M (Q'Q/2Vr/2-

Dividing bot h side s o f th e abov e inequalitie s b y \t — t'\al2, w e ge t

\at)w(x,t)-at')w(X,t')\\t~t'\-a/2

< \w\%' a/2)(2e)(a~a">'2 + 2 e - 1 ( 2 e ) 1 - a ' / 2 | w ; | ( Q ' a / 2 ) ( 2 e ) a / 2 | t r / 2

<C3e( Q-a ' ) / 2H£'a / 2 ) .

From thi s follow s (<»<b a ' / 2 ) < C 3 £ ( Q - Q ' ) / 2 | W | ^ ' Q / 2 ) . 4 .8 Give n a constan t C an d a n e > 0 , se t

fl = e - ( c + 1 ) t u , Q = Mx[0,T-e].

SOLUTIONS T O EXERCIS E PROBLEM S 199

From th e definition , th e sign s o f u an d u ar e th e same . The y satisfy , in M x (O.T) , th e inequalit y

dtu < Au — u.

Since u i s a continuou s functio n i n Q 1 ther e i s a poin t {x° ,t°) G Q where i t assume s th e maximu m value . I t suffice s t o deriv e a contradic -tion assumin g u(x° ,t°) > 0 . Sinc e w(-,0 ) < 0 fro m th e assumptio n o f u, w e mus t hav e t° > 0 . We readil y se e tha t thi s contradict s th e abov e inequality regardin g u, applyin g a simila r argumen t t o th e proo f o f Lemma 4.11 . Sinc e e i s arbitrary , w e ge t th e desire d conclusion .

4 .9 Conside r M t o b e a Riemannia n manifol d wit h an y Rie -mannian metric . Sinc e M i s compact , w e ca n choos e a finite cove r of M consistin g o f geodesi c sphere s B r(x\),... ,B r(xk) suc h tha t f(B3r(xi)) i s containe d i n a coordinat e neighborhoo d o f T V for eac h i = 1 , . . . , k. I n B^ r(xi) an d fo r sufficientl y smal l e , defin e

A W = / S / (47rt)~^ 2exp(-d(x,y)2/At)f(y)d^g(y)dt JO JB 2r(x1)

Choose a C°° functio n cpi : M — • M such tha t

0 < ifi < 1 , ipi(x) = 1 (x e B r(xi)), (fi(x) = 0 (x 0 B 3r/2(xi)).

Set / i = (pifi-\-(l — <pi)f. Then / i become s a functio n define d i n th e entire M . W e inductivel y defin e fi b y

fi(x) = [* [ ( 4 ^ ) - m / 2 e x p ( - d ( x , 7 / ) 2 / 4 0 / z - i ( y ) ^ ( l / ) ^ ,

and se t / ^ = ^ / i + ( 1 — (fi)fi_1. f k give s th e desire d C°° ma p / . I t

is readil y verifie d tha t / i s free-homotopi c t o / a s s — > 0 . 4 .10 A n elemen t a i n ^ ( M ) i s nothin g bu t th e homotop y clas s

of a continuou s ma p / : S k — > M fro m th e /c-dimensiona l spher e Sk int o M. Sinc e KM < 0 , / i s free-homotopi c t o a harmoni c ma p u : Sk — > M b y Corollar y 4.18 . Propositio n 4.24 , then , implie s tha t u is a constan t ma p fo r k > 2 ; hence , th e desire d conclusio n i s obtained .

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Bibliography

[I] J . Eells , Jr . an d J . H . Sampson , Harmoni c mapping s o f Riemannia n mani -folds, Amer. J. Math., 8 6 (1964) , 109-160 .

[2] J . Eells , Jr . an d J . C . Wood , Restriction s o f harmoni c amps , Topology, 15(1976), 263-266 .

[3] D . Gilbar g an d N . S . Trudinger , Elliptic partial differential equations of sec-ond order, Grundlehre n de r mathematische n Wissenschafte n 224 , 2n d edi -tion, Springer-Verlag , 1983 .

[4] R . S . Hamilton , Harmonic maps of manifolds with boundary, Lectur e Note s in Mathematic s 471 , Springer-Verlag , 1975 .

[5] P . Hartman , O n homotopi c harmoni c maps , Canad. J. Math., 19(1967) , 673-687.

[6] H . Hop f an d W . Rinow , Uebe r de n Begrif f de r vollstandige n differen t ialge-ometrischen Flachen , Comm. Math. Helv., 3(1931) , 209-225 .

[7] J . Kazdan , Some applications of partial differential equations to problems in geometry, Report s o n Globa l Analysis , VI , 1-130 , Semina r Kankokai , 1984 .

[8] W . Klingenberg , Lectures on closed geodesies, Grundlehle n de r mathematis -chen Wissenschafte n 230 , Springer-Verlag , 1978 .

[9] O . A . Ladyzenskaya , V . A . Solonnikov an d N . H. Ural'ceva , Linear and quasi-linear equations of parabolic type, Translation s o f Mathematica l Monograph s 23, Amer . Math.Soc , 1968 .

[10] S . Lang , Introduction to differentiable manifolds, Interscience , 1962 . [II] T . Levi-Civita , Notion e d i parallelisim o i n un a variet a qualunqu e e conse -

quente specificazion e geometric a dell a curvatur a Riemanniana , Rend. Circ. Mat. Palermo, 42(1917) , 173-204 .

[12] J . Milnor , Morse Theory, Annal s o f Mathematics Studie s 51 , Princeton Univ . Press, 1963 .

[13] M . J . Micalle f an d j . D. Moore , Minima l two-sphere s an d th e topolog y o f manifolds wit h positiv e curvatur e o n totall y isotropi c tw o planes , Ann. of Math., 127(1988) , 199-227 .

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[16] S . Nishikaw a an d K . Shiga , O n th e holomorphi c equivalenc e o f bounde d do -mains i n complet e Kahle r manifold s o f nonpositiv e curvature , J. Math. Soc. Japan, 35(1983 ) 273-278 .

[17] R . S . Palais , Mors e Theor y o n Hilber t manifolds , Topology, 2(1963) , 299-340 .

201

202 BIBLIOGRAPH Y

[18] R . S . Palai s an d S . Smale , A generalize d Mors e Theory , Bull Amer. Math, Soc, 70(1964) , 165-172 .

[19] A Preissmann , Quelque s propriet e globale s de s espace s d e Riemann , Coram. Math. Helv., 15(1943) , 175-21 6

[20] J . Sack s and K . Uhlenbeck , Th e existence of minimal immersion s o f 2-spheres , Ann. of Math., 113(1981) , 1-24 .

[21] S . Smale , Mors e Theory an d a non-linear generalizatio n o f the Dirichle t prob -lem, Ann. of Math., 80(1964) , 382-292 .

[22] J . T . Schwarz , Nonlinear functional analysis, Gordo n an d Breach , 1969 . [23] Y.-T . Siu , Th e complex-analyticit y o f harmoni c map s an d th e stron g rigidit y

of compac t Kahle r manifolds , Ann. of Math., 112(1980) , 73-111 : Stron g rigidity o f compac t quotient s o f exceptiona l bounde d symmetri c domains , Duke Math. J., 48(1981) , 857-871 .

[24] Y.-T . Si u an d S.-T . Yau , Compac t Kahle r manifold s o f positiv e bisectiona l curvature, Invent. Math., 59(1980) , 857-871 .

[25] J . L . Synge , O n th e connectivit y o f space s o f positiv e curvature , Quart. J. Math., 7(1936) , 316-320 .

[26] A . Treibergs , Lectures on the E'ells-Sampson theorem on parabolic deforma-tion of maps to harmonic maps, Universit y o f Uta h notes , 198 7

[27] S . Kobayashi , Complex geometry 1, Iwanam i koz a "Genda i sugak u n o kiso", Iwanam i Shoten , 1997 .

[28] Y . Matsumoto , Foundations of Morse Theory, Iwanam i koz a "Genda i sugak u no kiso" , Iwanam i Shoten , 1997 .

[29] M . Murat a an d K . Kurata , Partial differential equations 1, Iwanam i koz a "Gendai sugak u n o kiso" , Iwanam i Shoten , 1997 .

[30] T . Nagano , The global variational method, Kyorits u koza , "Genda i sugak u no kiso" , Kyorits u Shuppa n Sha , 1991 .

[31] H . Urakawa , The variational method and harmonic maps, Shokabo , 1990 .

BIBLIOGRAPHY 203

Books

1. T . Aubin , Nonlinear analysis on manifolds. Monge-Ampere equations, Springer-Verlag , 1982 .

This i s a seriou s introductor y boo k t o th e nonlinea r problem s that appea r i n differentia l geometry . Thi s boo k ma y b e regarde d as a treatis e o n nonlinea r analysi s o f Riemannia n manifold s suitabl e for th e reade r wh o ha s learne d analysis . I n particular , th e "Yamab e problem" an d th e "Calab i conjecture " ar e discusse d i n detail .

2. J . Eell s an d L . Lamaire , A repor t o n harmoni c maps , Bull. London Math. Soc, 10(1978) , 1-68 ; Selected topics in harmonic maps, C . B . M. S . Regional Conferenc e Serie s 50 , Amer. Math. Soc, 1983; Anothe r repor t o n harmoni c maps , Bull. London Math. Soc, 20(1988), 385-524 .

These thre e ma y b e regarde d a s a comprehensiv e repor t o n har -monic maps. The y are best suite d to survey the history an d the lates t developments i n th e stud y o f harmonic maps .

3. R . S . Hamilton , Harmonic maps of manifolds with boundary, Lecture Note s i n Mathematic s 471 , Springer-Verlag, 1975 .

This present s a solution t o the Dirichle t boundar y valu e proble m and th e Neuman n boundar y valu e problem unde r th e same curvatur e condition a s the theore m o f Eell s an d Sampson .

4. L . Jost , Harmonic mappings between Riemannian manifolds, Proceedings o f th e Centr e fo r Mathematica l Analysi s 4 , Australia n National Univ. , 1983 ; Nonlinear method in Riemannian and Kahler-ian geometry, DM V Semina r 1 0 Birkhauser, 1986 .

These articles discuss the differentiability o f weak solutions to th e equation o f harmoni c map s an d th e hea t flo w metho d regardin g th e Yang-Mills connection .

5. R . Schoe n an d S.-T . Yau , Lectures on harmonic maps, Inter -national Press , 1997 .

This contain s a serie s o f lecture s o n th e existenc e proble m o f harmonic map s betwee n Rieman n surfaces , th e application s o f th e theory o f harmoni c map s t o th e "topologica l spher e theorem " an d the "Franke l conjecture" , an d th e existenc e o f harmoni c map s int o spaces wit h singula r points , etc .

6. R . Schoe n an d S.-T . Yau , Lectures on differential geometry, International Press , 1994 .

204 BIBLIOGRAPHY

This i s a seriou s introductor y boo k t o th e stud y o f differentia l ge -ometry usin g analyti c methods . Thi s consist s o f a serie s o f lectures . The authors , fro m thei r ow n poin t o f view , discus s th e nonlinea r anal -ysis o n Riemannia n manifolds , usin g th e nonlinea r partia l differentia l equations tha t appea r i n problem s i n differentia l geometry . Ther e i s a collection o f "unsolve d problem s i n differentia l geometry " a t th e en d of th e book .

7. T . Ochiai , Editor , Nonlinear problems in differential geometry, Reports o n Globa l Analysi s I , Semina r Kankokai , 1979 ; S . Nishikaw a and T . Ochiai , Existence and applications of harmonic maps, Report s on Globa l Analysi s II , Semina r Kankokai , 1980 .

Discussed i n I ar e th e "Kazdan-Warne r problem" , th e "Yamab e problem", th e "Minkowsk i problem" , th e " Calab i conjecture" , etc . II contain s a treatis e o n th e "Platea u problem " regardin g minima l surfaces, an d a proo f fo r th e theore m o f Eell s an d Sampso n usin g estimates i n th e Sovole v spac e W k,p.

8. T . Ochiai , Editor , Minimal surfaces, Report s o n Globa l Anal -ysis IV , Semina r Kankokai , 1982 .

This consist s o f th e differentia l geometri c aspect s o f minima l sur -faces (Volum e 1) , application s o f minima l surface s (Volum e 2 ) an d the analytic s aspect s o f minima l surface s (Volum e 3) . Volum e 3 con -tains a detaile d discussio n o f th e solutio n o f Douglas-Rado-Morre y regarding th e existenc e o f minima l surface s an d it s properties .

9. T . Ochiai , Editor , Minimal surfaces, Report s o n Globa l Anal -ysis VI , XII , Semina r Kankokai , 1984 , 1989 .

These report s contai n paper s b y J . L . Kazdan , Som e applica -tions o f partia l differentia l equation s t o problem s i n geometry ; S . Nishikawa, O n continuit y o f wea k solution s t o non-linea r ellipti c par -tial differentia l equations , I , II ; A . Tachikawa , O n differentiabilit y o f the solution s t o variationa l problems , etc . Th e lectur e note s o f Kaz -dan ar e bes t suite d a s a n introductio n t o thes e topics .

10. T . Sakai , Riemannian Geometry, Shokabo , 1992 .

This boo k contain s detaile d account s o n th e relationshi p betwee n the curvatur e an d topolog y o f Riemannia n manifolds . Specia l at ten -tion i s pai d t o th e metho d calle d th e compariso n theore m an d it s application.

11. H . Urakawa , The variational method and harmonic maps, Shokabo, 1990 .

BIBLIOGRAPHY 205

This wel l writte n boo k explain s th e genera l theor y o f th e varia -tional metho d an d th e theor y o f harmonic map s startin g fro m a ver y basic level . I n particular , i t contain s Uhlenbeck' s proo f o f th e theo -rem o f Eells an d Sampso n fro m th e poin t o f view o f Morse theor y o n infinite dimensiona l manifolds .

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Index

a Holde r continuou s function , 17 5 c°° curve , 6 C°° vecto r field , 1 7 C°° coordinat e neighborhoo d

system, 16 4 C°° diffeomorphic , 16 7 C°° diffeomorphism , 16 6 C°° function , 16 2 C°° manifolds , 16 2 C°° map , 16 5 C°° section , 64 , 17 2 C°° tenso r field , 5 5 C°° variation , 48 , 9 9 C°° vecto r field , 17 0 (r, s) tensor , 5 4 (r, s) tenso r bundle , 6 4 (r, s) tenso r field , 5 5

action integral , 8 , 8 9 afflne connection , 1 9 afflne parameter , 2 8 antiholomorphic function , 11 0 arc length , 1 3 area, 11 8

Banach space , 16 3 Bianchi firs t identity , 5 8 Bianchi secon d identity , 11 7 bounded linea r operator , 17 4 bracket product , 1 8 bundle o f homomorphisms , 17 3

characteristic polynomial , 17 6 Christoffel symbols , 1 5 closed geodesic , 7 4 commutator product , 1 8 compatible, 2 5

complete, 7 0 component, 3 , 4 , 17 , 54 , 166 , 17 0 conformal, 11 8 connection coefficient , 2 0 coordinate function , 16 4 coordinate neighborhood , 16 4 cotangent bundle , 64 , 16 9 covariant derivative , 21 , 91, 11 1 covariant derivativ e i n th e directio n

of v , 2 1 covariant differential , 9 1 covariant differentiatio n alon g c , 2 3 critical point , 1 3 critical value , 1 3 curvature tensor , 5 8

derivative, 16 6 diffeomorphism, 16 6 dimension, 17 1 direct sum , 17 3 Dirichlet integral , 10 6 Dirichlet principle , 10 6 distance, 4 2 divergence, 11 5 dual vecto r bundle , 17 3

elliptic, 17 6 energy, 8 , 8 9 energy density , 8 9 equation fo r harmoni c maps , 10 4 Euler's equation , 1 3 exponential map , 3 2

fiber, 30 , 64 , 108 , 17 1 fiber metric , 17 2 first variatio n formula , 1 3

208 INDEX

Frechet differentiable , 17 4 free homotopy , 7 4 free homotop y class , 7 4 functional, 1 3 fundamental lemm a i n th e theor y

of variation , 4 4 fundamental solution , 17 7

Gateaux differen t iable, 17 4 Gauss lemma , 4 0 geodesic, 15 , 2 7 geodesic ball , 3 9 geodesic flow , 4 5 geodesic loop , 7 4 geodesic sphere , 3 9 geodesic spray , 4 5 geodesically complete , 7 0 gradient, 11 5 Green's theorem , 11 6

Holder continuou s function , 17 5 Holder space , 13 6 harmonic function , 10 6 harmonic map , 10 4 heat equation , 17 7 heat flo w method , 11 9 heat operator , 17 7 Hilbert space , 17 4 holomorphic map , 11 0 homomorphism, 17 1 Hopf map , 11 7 horizontal component , 10 8 horizontal lift , 10 8

imbedded submanifold , 16 7 imbedding, 16 7 immersed submanifold , 16 7 immersion, 16 7 induced connection , 11 1 induced metric , 5 induced vecto r bundle , 17 3 integral curve , 17 0 isometric, 8 3 isometric immersion , 10 7 isomorphism, 17 2

Laplace operator , 11 5 Laplacian, 11 5 length, 7

Levi-Civita connection , 2 5 Lie derivative , 4 5 linear connection , 1 9 linear partia l differentia l operato i

176 local coordinat e system , 16 4 local coordinates , 16 4 local on e paramete r transformatio i

group, 17 1 local triviality , 17 1 locally finite, 16 8 loop, 7 4

maximal principle , 14 2 minimal submanifold , 10 7 minimizing sequence , 7 6

natural frame , 1 7 norm, 17 3 normal coordinat e neighborhood , 3 normal coordinat e system , 3 7 normed space , 17 3

one paramete r transformatio n group,171

open submanifold , 16 4

parabolic, 17 6 parabolic equatio n fo r harmoni c

maps, 12 3 paracompact, 16 8 parallel, 23 , 9 2 parallel displacement , 2 4 parameter, 6 partition o f unity , 16 8 piecewise smoot h curve , 6 piecewise smoot h variation , 4 7 Poincare uppe r half-space , 3 4 product manifold , 16 4 projection, 30 , 169 , 17 1

rank, 17 1 real hyperboli c space , 3 4 refinement, 16 8 regular, 7 Ricci curvature , 6 2 Ricci identity , 124 , 16 1 Ricci tensor , 6 2 Riemannian connection , 2 5 Riemannian curvatur e tensor , 5 9

Riemannian metric , 4 , 17 2 Riemannian submersions , 10 7

scalar curvature , 6 3 Schauder estimate , 18 1 second fundamenta l form , 9 6 sectional curvature , 6 1 smooth curve , 6 smooth manifold , 16 3 smooth variation , 4 8 space o f constan t curvature , 6 2 standard connection , 1 9 standard measure , 11 4 submanifold, 16 7 submersion, 10 7 support, 16 8 symmetric, 2 5

tangent bundle , 30 , 164 , 16 9 tangent space , 16 5 tangent vector , 16 5 tangent vecto r field, 2 2 tension field, 9 8 tensor field, 9 7 tensor product , 4 , 17 3 totally geodesic , 16 2 trivial, 7 4 tubular neighborhood , 16 1

unit spee d geodesic , 2 8

variational vecto r field, 48 , 9 9 vector bundle , 17 1 vector field, 1 7 vector field alon g u 1 3 8 vector subbundle , 17 2 vertical component , 10 9

Weitzenbock formula , 12 4

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Titles i n Thi s Serie s

205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 2

204 A . M . Vinogradov , Cohomologica l analysi s o f partia l differentia l

equations an d Secondar y Calculus , 200 1

203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d

coding, 200 2

202 V . P . Maslo v an d G . A . Omel ' yanov , Geometri c asymptotic s fo r

nonlinear PDE . I , 200 1

201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1

200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1

199 Shigeyuk i Morita , Geometr y o f characteristi c classes , 200 1

198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology ,

2001

197 Kenj i Ueno , Algebrai c geometr y 2 : Sheave s an d cohomology , 200 1

196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes ,

2001

195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1

194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1

193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1

192 Yu . P . Solovyo v an d E . V . Troitsky , C*-algebra s an d ellipti c

operators i n differentia l topology , 200 1

191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n

geometry, 200 0

190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces ,

2000

189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e

phenomena, 200 0

188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f

random variable s an d rando m processes , 200 0

187 A . V . Fursikov , Optima l contro l o f distribute d systems . Theor y an d

applications, 200 0

186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r

theory 1 : Fermat' s dream , 200 0

185 Kenj i Ueno , Algebrai c Geometr y 1 : From algebrai c varietie s t o schemes ,

1999

184 A . V . Mel'nikov , Financia l markets , 199 9

183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9

182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d

conservation law s fo r differentia l equation s o f mathematica l physics , 199 9

181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups .

Part 2 , 199 9

180 A . A . Milyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d optimal control , 199 8

TITLES I N THI S SERIE S

179 V . E . Voskresenskii , Algebrai c group s an d thei r birationa l invariants , 1998

178 Mitsu o Morimoto , Analyti c functional s o n th e sphere , 199 8

177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8

176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e

Hamiltonian system s wit h simpl e singula r point s (topologica l aspects) , 199 8

175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8

174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d

elliptic systems , 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l

properties o f distribution s o f stochasti c functionals , 199 8

172 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups .

Part 1 , 199 8

171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8

170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c

integrals, 199 7

169 S . K . Godunov , Ordinar y differentia l equation s wit h constan t

coefficient, 199 7

168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8

167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s

of fractals , 199 7

166 Kenj i Ueno , A n introductio n t o algebrai c geometry , 199 7

165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g

problem i n Galoi s theory , 199 7

164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis ,

1997

163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I .

Yadrenko, Probabilit y theory : Collectio n o f problems , 199 7

162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d

methods i n linea r statistica l models , 199 7

161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c

dynamics, 199 7

160 V . G . OsmolovskiT , Linea r an d nonlinea r perturbation s o f th e operato r

div, 199 7

159 S . Ya . Khavinson , Bes t approximatio n b y linea r superposition s

(approximate nomography) , 199 7

158 Hidek i Omori , Infinite-dimensiona l Li e groups , 199 7

157 V . B . Kolmanovski i an d L . E . ShaTkhet , Contro l o f system s wit h aftereffect, 199 6

For a complet e lis t o f title s i n thi s series , visi t th e AMS Bookstor e a t www.ams.org/bookstore/ .