Selected Title s i n Thi s Serie s

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 U e n o , 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 . Buldyg i 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 , Nobush ig e Kurokawa , an d Takesh i Saito , Numbe r

theory 1 : Fermat' s dream , 200 0

185 Kenj i U e n o , 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 . Mi lyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d

optimal control , 199 8

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

1998

178 Mi tsu o Mor imoto , 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

(Continued in the back of this publication)

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Geometry of Differential Form s

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 20 1

Geometry o f Differential Form s

Shigeyuki Morit a

Translated b y

Teruko Nagas e Katsumi Nomiz u

^ ^ „JSfr Wi^^fm America n Mathematica l Societ y "* " ih Providence , Rhod e Islan d

10.1090/mmono/201

Editor ia l Boar d

Shoshichi Kobayash i (Chair ) Masamichi Takesak i

wttw^omm^ 1,2 B I B U N K E I S H I K I N O K I K A G A K U

( G E O M E T R Y O F D I F F E R E N T I A L F O R M S )

b y S h i g e y u k i M o r i t a

Copyright © 1997 , 199 8 b y Shigeyuk i Mori t a Originally publ ishe d i n Japanes e

by Iwanam i Shoten , Publ ishers , Tokyo , 1997,199 8

Translated fro m th e Japanes e b y Teruk o Nagas e an d Ka t sum i Nomiz u

2000 Mathematics Subject Classification. P r imar y 57Rxx , 58Axx .

ABSTRACT. Thi s boo k i s a comprehensiv e introductio n t o differentia l forms . I t begins wit h a quic k introductio n t o th e notio n o f differentiat e manifolds , an d then develop s basi c propertie s o f differentia l form s a s wel l a s fundamenta l result s concerning them , suc h a s th e d e Rha m an d Frobeniu s theorems . Th e secon d half o f th e boo k i s devoted t o mor e advance d material , includin g Laplacian s an d harmonic form s o n manifolds , th e concept s o f vecto r bundle s an d fibe r bundles , and th e theor y o f characteristi c classes . Amon g th e les s traditiona l topic s treate d is a detaile d descriptio n o f th e Chern-Wei l theory .

The boo k ca n serv e a s a textboo k fo r a n undergraduat e o r graduat e cours e i n geometry.

L i b r a r y o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a

Morita. S . (Shigeyuki) , 1946 -[Bibun keishik i n o kikagaku . English ] Geometry o f differentia l form s / Shigeyuk i Morit a ; translated b y Teruk o Na -

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

v 201 ) (Iwanami serie s i n moder n mathematics ) Includes bibliographica l reference s an d index . ISBN 0-8218-1045- 6 (softcove r : alk . paper ) 1. Differentia l forms . 2 . Differentiabl e manifolds . I . Title . II . Series .

III. Series : Iwanam i serie s i n moder n mathematics .

QA381.M67 200 1 515'. 37—dc21 2001022608

© 200 1 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

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

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Information o n copyin g an d reprintin g ca n b e foun d i n th e bac k o f thi s volume . Visit th e AM S hom e pag e a t URL : http:/ /www.ams.org /

10 9 8 7 6 5 4 32 1 5 14 13 12 11 10

Contents

Preface xin

Preface t o th e Englis h Editio n

Outline an d Goa l o f th e Theor y

Chapter 1 Manifold s 1.1 Wha t i s a manifold ?

(a) Th e n-dimensiona l numerica l spac e E n

(b) Topolog y o f R n

(c) C°° function s an d diffeomorphism s (d) Tangen t vector s an d tangen t space s o f R n

(e) Necessit y o f a n abstrac t definitio n Definition an d example s o f manifold s

(a) Loca l coordinate s an d topologica l manifold s (b) Definitio n o f differentiabl e manifold s (c) W 1 an d genera l surface s i n i t (d) Submanifold s (e) Projectiv e space s (f) Li e group s

1.2

(a (b

(o

(e

(f

(a (b

1.3 Tangen t vector s an d tangen t space s C°° function s an d C°° mapping s o n manifold s Practical constructio n o f C°° function s o n a man -ifold Partition o f unit y Tangent vector s The differentia l o f map s Immersions an d embedding s

1.4 Vecto r field s Vector fields The bracke t o f vecto r fields

xvn

xix

1 2 2 3 4 6 10 11 11 13 16 19 21 22 23 23

25 27 29 33 34 36 36 38

viii CONTENT S

(c) Integra l curve s o f vecto r fields an d one-paramete r group o f loca l transformation s 3 9

(d) Transformation s ofvecto r field s by diffeomorphism 4 4 1.5 Fundamenta l fact s concernin g manifold s 4 4

(a) Manifold s wit h boundar y 4 4 (b) Orientatio n o f a manifol d 4 6 (c) Grou p action s 4 9 (d) Fundamenta l group s an d coverin g manifold s 5 1

Summary 5 4 Exercises 5 5

Chapter 2 Differential Form s 5 7 2.1 Definitio n o f differentia l form s 5 7

(a) Differentia l form s o n R n 5 7 (b) Differentia l form s o n a genera l manifol d 6 1 (c) Th e exterio r algebr a 6 1 (d) Variou s definition s o f differentia l form s 6 6

2.2 Variou s operation s o n differentia l form s 6 9 (a) Exterio r produc t 6 9 (b) Exterio r differentiatio n 7 0 (c) Pullbac k b y a ma p 7 2 (d) Interio r produc t an d Li e derivative 7 2 (e) Th e Carta n formul a an d propertie s o f Li e deriva -

tives 7 3 (f) Li e derivativ e an d one-paramete r grou p o f loca l

transformations 7 7 2.3 Frobeniu s theore m 8 0

(a) Frobeniu s theore m — Representatio n b y vecto r fields 8 0

(b) Commutativ e vecto r fields 8 2 (c) Proo f o f the Frobeniu s theore m 8 3 (d) Th e Frobenius theorem Representatio n b y dif-

ferential form s 8 6 2.4 A few fact s 8 9

(a) Differentia l form s wit h value s i n a vecto r spac e 8 9 (b) Th e Maurer-Carta n for m o f a Li e group 9 0

Summary 9 2 Exercises 9 3

Chapter 3 The d e Rha m Theore m 3.1 Homolog y o f manifold s

95 96

CONTENTS

(b (c (d

3.2 Integra l o f differentia l form s an d th e Stoke s theore m (a (b (c

(a; (b (c

(b

(c (d

(a (b (c (d (e

Summary Exercises

Homology o f simplicia l complexe s Singular homolog y C°° triangulatio n o f C° ° manifold s C°° singula r chai n complexe s o f C^ manifolds

Integral o f n-form s o n n-dimensiona l manifold s The Stoke s theore m (i n th e cas e o f manifolds ) Integral o f differentia l form s o n chains , an d th e Stokes theore m

3.3 Th e d e Rha m theore m de Rha m cohomolog y The d e Rha m theore m Poincare lemm a

3.4 Proo f o f th e d e Rha m theore m Cech cohomolog y Comparison o f de Rham cohomolog y an d Cec h co-homology Proof o f th e d e Rha m theore m The d e Rha m theore m an d produc t structur e

3.5 Application s o f th e d e Rha m theore m Hopf invarian t The Masse y produc t Cohomology o f compac t Li e group s Mapping degre e Integral expression of the linking number by Gaus s

Chapter 4 Laplacian an d Harmoni c Form s 4.1 Differentia l form s o n Riemannia n manifold s

(a) Riemannia n metri c (b) Riemannia n metri c an d differentiea l form s (c) Th e *-operato r o f Hodg e

4.2 Laplacia n an d harmoni c form s 4.3 Th e Hodg e theore m

(a) Th e Hodg e theore m an d th e Hodg e decomoposi -tion o f differentia l form s

(b) Th e ide a fo r th e proo f o f th e Hodog e decomposi -tion

4.4 Application s o f th e Hodg e theore m

96 99 100 103 104 104 107

109 111 111 113 116 119 119

121 126 131 133 133 136 137 138 140 142 142

145 145 145 148 150 153 158

158

160 162

CONTENTS

(a) Th e Poincar e dualit y theore m (b) Manifold s an d Eule r numbe r (c) Intersectio n numbe r

Summary Exercises

Chapter 5 Vector Bundle s an d Characteristi c Classe s 5.1 Vecto r bundle s

(a: (b; (c

The tangen t bundl e o f a manifol d Vector bundle s Several construction s o f vector bundle s

5.2 Geodesie s an d paralle l translatio n o f vector s (a) Geodesie s (b) Covarian t derivativ e (c) Paralle l displacemen t o f vectors an d curvatur e

5.3 Connection s i n vecto r bundle s an d (a) Connection s (b) Curvatur e (c) Connectio n for m an d curvatur e for m (d) Transformatio n rule s o f the loca l expressions fo r a

connection an d it s curvatur e (e) Differentia l form s wit h value s i n a vecto r bundl e

5.4 Pontrjagi n classe s (a) Invarian t polynomial s (b) Definitio n o f Pontrjagin classe s (c) Levi-Civit a connectio n

5.5 Cher n classe s (a) Connectio n and curvature in a complex vector bun-

dle (b) Definitio n o f Cher n classe s (c) Whitne y formul a (d) Relation s betwee n Pontrjagi n an d Cher n classe s

5.6 Eule r classe s (a) Orientatio n o f vector bundle s (b) Th e definitio n o f th e Eule r clas s (c) Propertie s o f the Eule r clas s

5.7 Application s o f characteristic classe s (a) Th e Gauss-Bonne t theore m (b) Characteristi c classe s o f th e comple x projectiv e

space

162 164 165 166 167

169 169 169 170 173 180 180 181 183 185 185 186 188

190 191 193 193 197 201 204

204 205 207 208 211 211 211 214 216 216

223

CONTENTS

(c) Characteristi c number s Summary Exercises

Chapter 6 Fiber Bundle s an d Characteristi c Classe s 6.1 Fibe r bundl e an d principa l bundl e

(a) Fibe r bundl e (b) Structur e grou p (c) Principa l bundl e (d) Th e classificatio n o f fiber bundle s an d character -

istic classe s (e) Example s o f fibe r bundle s

6.2 S 1 bundle s an d Eule r classe s (a) S l bundl e (b) Eule r clas s o f a n S 1 bundl e (c) Th e classificatio n o f S l bundle s (d) Definin g th e Eule r clas s for a n S 1 bundl e by usin g

differential form s (e) Th e primar y obstructio n clas s and th e Eule r clas s

of th e spher e bundl e (f) Vecto r field s o n a manifol d an d Hop f inde x theo -

rem 6.3 Connection s

(a) Connection s i n genera l fibe r bundle s (b) Connection s i n principa l bundle s (c) Differentia l for m representatio n o f a connection i n

a principa l bundl e 6.4 Curvatur e

(a) Curvatur e for m (b) Wei l algebr a (c) Exterio r differentiatio n o f the Wei l algebr a

6.5 Characteristi c classe s Weil homomorphis m Invariant polynomial s fo r Li e group s Connections fo r vecto r bundle s and principa l bun -dles Characterisric classe s

(a

(b

(a (b (c.

(d: 6.6 A coupl e o f item s

Triviality o f th e cohomolog y o f th e Wei l algebr a Chern-Simons form s

225 228 229

231 231 231 233 236

238 239 240 241 241 246

249

254

255 257 257 260

262 265 265 268 270 275 275 279

282 284 285 285 287

CONTENTS

(c) Fla t bundle s an d holonom y homomorphism s Summary Exercises

Perspectives

Answers to Exercise s Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6

287 291 292

295

299 299 302 305 308 310 311

References 31 5

Index 317

Preface

As th e titl e indicates , thi s boo k i s a n expositio n o f differentia l forms. Wha t i s a differential form ? Th e aim of this book i s to answe r that question .

To explain differentia l forms , we have to comment on the differen -tiable manifolds ove r which they ar e defined. I n brief, a differentiabl e manifold i s a moder n representatio n o f a figure a s a geometri c ob -ject, an d i s a n importan t notio n i n moder n mathematics . Therefor e many textbooks on differentiable manifold s hav e been published. Th e reader ma y hav e seen or eve n alread y studie d som e of them. I n thes e textbooks, without exception , differentia l form s are defined. However , in many case s only th e definitio n an d th e fundamenta l propertie s ar e presented, whil e only a brief descriptio n i s given of how they ar e use d to analyze the structure o f the differentiable manifold . Thi s is because so many fundamenta l fact s alread y tak e u p a lo t o f space .

Now tha t th e notio n o f differentiabl e manifol d i s completely es -tablished, thi s situatio n i s i n a sens e inevitable . However , a n in -convenience arises . Tha t is , th e reade r wil l b e bus y studyin g th e fundamental fact s o f differentiabl e manifolds , an d wil l b e lef t wit h little tim e fo r practica l manifolds . Also , as a theoretically reasonabl e description i s frequentl y no t i n th e orde r o f historica l development , the excitemen t o f th e discover y ofte n get s lost .

Modern mathematic s i s no w progressin g dynamically . I n geom -etry a revolutionar y chang e starte d i n th e 1980s , an d i t continue s today wit h n o sig n o f halting . I n suc h a n er a o f progress , i t i s ver y important t o understand mathematic s a s a living system that i s ready for ne w advances , rathe r tha n a s a completely establishe d system .

In thi s serie s ther e i s n o boo k entitle d "Manifolds" . Thi s ma y be becaus e th e serie s editor s too k th e abov e fact s int o account , an d desired to lead the readers to the scene of current activ e mathematica l research a s quickl y a s possible .

xii i

xiv P R E F A C E

When I started t o write thi s book , I found i t muc h mor e difficul t than I had expected . Sinc e mathematic s stand s o n logic , vagu e de -scriptions ar e no t allowed. O n th e othe r hand , i f I trie d t o bas e m y explanations o n th e historica l motivations , th e boo k woul d quickl y grow too long. I have tried t o find a reasonable compromise , an d wil l leave i t t o th e reade r t o judge ho w clos e I have com e t o th e goal .

The contents of this book can be summarized a s follows. I n Chap-ter 1 we begin wit h th e definitio n o f differentiabl e manifold s an d th e fundamental idea s connected wit h them, suc h as tangent vectors , tan -gent spaces , etc . Th e description , althoug h minimal , shoul d b e suffi -cient fo r understandin g th e res t o f thi s book .

In Chapte r 2 we introduc e differentia l forms , defin e thei r funda -mental operations , an d the n prov e th e theore m o f Frobenius . Thi s theorem gives a necessary and sufficien t conditio n fo r the integrabilit y of "field s o f directions" give n a t ever y poin t o f a manifold , whic h ar e described b y eithe r differentia l form s o r vecto r fields, an d it s impor -tance ha s been increasin g recently .

The them e o f Chapte r 3 is the theore m o f d e Rham . Th e reade r may hav e hear d th e nam e o f thi s theorem . I n fac t i t i s a ver y im -portant result , an d w e may even sa y tha t i t serve s a s the basi s o f th e theory of manifolds. I n addition t o the ordinary proof , w e give an ex -planation to clarify it s relation to the integration o f differential forms . Also, severa l application s o f thi s theore m ar e give n a t th e en d o f th e chapter. Althoug h the y ma y b e somewha t difficult , th e autho r hope s that the y wil l show th e reade r som e o f the powe r o f thi s theorem .

The secon d par t o f thi s boo k begin s i n Chapte r 4 , i n whic h w e study th e relationshi p betwee n Riemannia n metric s an d differentia l forms. W e then explai n th e beautifu l theor y o f harmoni c forms , du e to Hodge and als o to Kodair a an d d e Rham. Briefly , thi s theory ma y be sai d t o giv e a refinemen t o f d e Rham' s theore m i n th e contex t o f Riemannian manifolds .

In Chapte r 5 , we introduce th e notio n o f vecto r bundles . Thi s i s the notion obtained b y generalizing th e tangen t bundl e of a manifold , and i t i s a crucia l too l i n moder n mathematics . W e als o explai n th e concepts of connection an d curvature , which are used t o measure ho w vector bundle s ar e twisted .

In th e las t chapter , Chapte r 6 , w e explai n th e theor y o f charac -teristic classes , whic h I woul d sa y i s on e o f th e highes t summit s o f modern geometry . B y virtu e o f thi s theory , th e structur e o f figures, namely manifolds , ca n b e expresse d i n term s o f differentia l forms ,

PREFACE xv

which are local objects; an d i f we integrate them, the global structur e appears a s concret e numbers , calle d characteristi c numbers . Her e almost al l o f the previou s materia l wil l be used .

Those reader s wh o wan t t o kno w th e detail s o f manifolds o r ho -mology theor y tha t ar e use d i n thi s boo k ar e invite d t o consul t text -books on those subjects. I f this book leads the reader to such a study, or i f i t awaken s a n interes t i n deeper theories , the autho r wil l be very happy indeed .

Shigeyuki Morit a July 199 6

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Preface t o th e Englis h Editio n

This i s a translation o f my boo k originall y publishe d i n Japanes e by Iwanam i Shoten , Publishers . I t aim s a t introducin g th e reade r t o the theor y an d practic e o f differentia l form s o n manifolds , assumin g only a minimu m o f knowledg e suc h a s linea r algebra , calculus , an d elementary topology . I t als o include s a quic k introductio n t o th e concept o f differentiable manifolds . I hope that thi s book wil l provide the reade r wit h a flavor o f moder n geometr y an d encourag e hi m o r her t o procee d t o a stud y o f deepe r theories .

The original Japanese edition was published i n two volumes. I am grateful t o Mrs. Teruk o Nagase for translating the first par t (Chapter s 1,2,3) whil e keepin g clos e contac t wit h th e autho r durin g th e work . The secon d par t (Chapter s 4,5,6 ) wa s translate d b y Professo r Kat -sumi Nomizu , wh o als o suggeste d severa l improvement s o n th e text . I woul d lik e t o expres s my dee p gratitud e t o him . Finally , I woul d like to than k th e America n Mathematica l Societ y fo r publishin g thi s English edition , an d thei r staf f fo r providin g excellen t support .

Shigeyuki Morit a January 200 1

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Outline an d Goa l o f th e Theor y

Geometry i s th e scienc e o f figures. W e stud y variou s propertie s of figures, an d classif y give n figures accordin g t o th e results . W e have th e notio n o f invariants , whic h ca n serv e a s th e mos t effectiv e method o f classification . W e ma y briefl y sa y tha t invariant s describ e geometric structure s i n term s o f numbers . Fo r example , a s i s wel l known, th e conditio n fo r congruenc e o f triangles i s described b y such invariants a s th e lengt h o f edge s an d angle s a t vertices .

However, i n geometr y w e ar e no t alway s studyin g give n figures. Sometimes w e ask wha t kin d o f figures can exis t a t all , and als o enu -merate condition s fo r thei r existence . I t ma y b e a n eterna l problem , both i n physics and i n mathematics, fo r human s to imagine the figure and shap e o f th e univers e wher e w e live , an d t o stud y thos e condi -tions. I n geometry , i n som e cases , w e ca n eve n surpris e peopl e b y constructing unknow n figures practically . Thi s i s one of the pleasure s of studyin g geometry . Th e appearanc e o f non-Euclidia n geometr y i s surely a typica l example , whic h show s tha t th e axio m o f parallel s i s not tru e i n general .

The figures tha t ar e treate d i n modern geometr y ar e calle d man -ifolds. I t i s usuall y sai d tha t th e notio n o f manifold s w ras introduce d by Rieman n i n hi s inaugura l lectur e a t Gottinge n Universit y i n 1854 . In thi s talk , th e geometr y o f manifold s wit h Riemannia n metric s -namely, differentia l geometr y - wa s also initiated. Thi s was an epoch-making lecture , i n advanc e o f it s time . W e als o ow e a grea t dea l t o a serie s o f work b y Whitney , begu n i n the 1930's , fo r the formulatio n of manifold s tha t w e are no w using .

Although the y ar e al l calle d "manifolds" , ther e ar e various kind s of manifolds . Th e simples t ar e th e topologica l manifolds , whic h w e only requir e t o b e locall y homeomorphi c t o a Euclidea n space . How -ever, a manifol d usuall y mean s a differentiabl e manifold , whic h ha s smoothness; example s includ e curve s an d surface s wit h beautifu l

XIX

XX OUTLINE AND GOAL OF THE THEORY

curved shapes . Ther e ar e als o complex manifold s an d algebrai c man -ifolds (o r varieties), which have finer structures . Nowaday s eac h kin d of manifol d i s studie d b y it s ow n methods . However , w e shoul d no t forget th e origin o f al l manifolds, bac k i n the day s when mathematic s was not separated int o branches like today. Tha t is , we must conside r those manifold s tha t wer e considere d b y ou r grea t predecessor s suc h as Gauss , Rieman n an d Poincare .

As a simpl e bu t ver y importan t example , le t u s conside r ori -entable close d surfaces . Whil e w e shal l giv e a n exac t definitio n late r on, fo r th e presen t jus t imagin e a smoot h surfac e i n spac e whic h i s bounded an d ha s n o boundary, a s i n Figur e 0.1 . Th e classificatio n o f

FIGURE 0.1 . Orientabl e close d surfac e

these surfaces wa s already completed a t th e beginning of this century . There i s an invarian t whic h come s t o min d fro m a first glanc e a t th e figure, namel y th e numbe r o f holes , whic h i s calle d th e genus . The n a necessar y an d sufficien t conditio n tha t tw o close d surface s b e th e same (i n moder n language , homeomorphi c o r diffeomorphic ) i s tha t their gener a are equal. S o if E 5 denote s a genus g closed surface , the n the infinit e sequenc e

So,Si ,E2, . . . . exhausts al l th e orientabl e close d surfaces . S o i s th e spher e an d E i is th e surfac e calle d a torus . Usuall y the y ar e denote d b y S 2 an d T2, respectively . I t i s no t a n exaggeratio n t o sa y tha t al l th e essenc e of geometry i s contained i n the abov e classificatio n o f closed surface s and i n th e Gauss-Bonne t theore m mentione d below . Actually , on e goal o f 20t h centur y geometr y wa s t o tr y t o exten d thes e fact s t o general manifold s i n highe r dimensions .

By the way, even if we consider only surfaces where the conclusio n is extremely simpl e an d clear , i f w e thin k abou t it s meanin g a littl e more deeply, we find that th e problem i s not s o simple. I t ma y appea r that th e meaning of the genus g is so obvious from the figure that ther e is n o difficult y i n definin g it . However , thi s i s onl y because , i n thi s picture, th e figure i s positioned i n such a way tha t it s genus is clearly

OUTLINE AN D GOA L O F T H E THEOR Y X X I

shown. I n the case of a complicated surface , i t i s impossible to be sure of th e genu s a t sight , n o matte r ho w smal l i t is . Moreover , ther e i s the intrinsicall y mor e importan t poin t tha t manifolds ar e no t alway s located in the well-known Euclidean spaces. I n fact, i t is characteristic of modern geometr y tha t manifold s ar e independent o f the framework of Euclidea n spaces , an d becam e quit e fre e objects . Therefore , whe n we study them , w e cannot alway s utiliz e thei r relativ e relatio n t o th e whole space . Moreover , i n th e case s o f higher dimensiona l manifolds , it i s impossibl e t o observ e the m directl y wit h ou r eyes , n o matte r how much w e try t o stretc h ou r imagination . S o how can we manag e them?

One possible method woul d be a combinatorial one where we take certain item s a s unit s an d decompos e manifold s int o these items . A s the items , w e ca n us e points , lines , triangles , an d wha t ar e calle d simplices, which ar e thei r generalization s t o general dimensions. Als o we coul d us e cells , whic h hav e mor e flexibl e shapes . Thi s metho d i s very practical, an d historicall y the firs t geometri c invariant , the Eule r number, wa s foun d b y thi s combinatoria l method . I f w e decompos e a give n figur e int o severa l triangle s an d tak e th e alternatin g su m o f the number s o f vertices , edge s an d triangles , the n th e tota l i s inde -pendent o f the metho d o f decomposition, an d i t constitute s a specifi c quantity o f figures . Man y reader s ma y kno w tha t i n th e cas e o f a genus g close d surface , i t i s equa l t o 2 — 2# , an d thi s fac t i n tur n indicates tha t th e genu s could b e denned b y a combinatorial method . This is homology theory , introduce d b y Poincare about 10 0 years ago, which extended thes e idea s and becam e a n importan t mean s to stud y figures b y combinatoria l methods . Thi s theor y enable s u s t o coun t the numbe r o f "holes " i n eac h dimensio n (called th e Bett i number) . Thus the Eule r numbe r wa s given a theoretical basi s for th e firs t tim e by Poincare , an d s o i t i s sometime s als o calle d th e Euler-Poincar e characteristic. I n th e 20t h century , cohomolog y group s wer e define d as the dua l o f homolog y groups , and , wit h bot h o f them, a branc h o f geometry calle d algebrai c topolog y flourished .

Another metho d originate d fro m th e theor y o f surface s du e t o Gauss, a s well as the Gauss-Bonne t theore m whic h followed it , an d i t uses differentiation an d integratio n t o study figures . Althoug h w e say simply a genu s g closed surface , ther e ar e various way s of realizing i t in th e space . I n mor e mathematica l terms , ther e ar e various kind s of Riemannian structure s o n S p , an d w e can ben d i t quit e freely . Wha t Gauss showed i n his theory o f surfaces i s that ho w curved a surface is ,

XX11 OUTLINE AND GOAL OF THE THEORY

now calle d th e curvature , i s a n intrinsi c quantit y o f th e surfac e an d can b e defined apar t fro m th e spac e wher e i t lies . Becaus e o f this , i t may b e sai d tha t h e se t th e stag e fo r th e above-mentione d wor k o f Riemann. Th e Gauss-Bonne t theore m claim s tha t i f we integrate th e curvature K o f a genus g closed surface 5 , whic h i s curved arbitrarily , over th e whol e surface , the n th e resul t i s a constan t independen t o f how it is curved, and that constan t i s 2n times the Euler number x(&)-If w e expres s thi s i n mathematica l terms , w e obtai n th e followin g beautiful equation :

f Kda = 2TTX(S).

Now i f w e tr y t o describ e th e goa l o f moder n geometr y i n on e sentence, w e may sa y tha t thi s goa l i s t o exten d th e classificatio n o f closed surface s an d als o th e Gauss-Bonne t theore m t o manifold s o f arbitrary dimensio n i n various ways . Her e differentia l form s playe d a fundamental role .

First o f all , th e theore m o f d e Rha m claim s tha t th e homolog y as wel l a s th e cohomolog y groups , whic h ar e define d b y combina -torial methods , ca n b e obtaine d usin g differentia l form s i n th e cas e of differentiabl e manifolds . Bu t then , ho w doe s i t go ? Element s i n the /c-dimensiona l homolog y grou p o f a manifol d ar e represente d b y so-called fc-dimensional cycles . A cycl e i s literall y a figure withou t boundary whic h return s t o itself . I n the case s where k = 0,1,2 , a cy -cle may be understood t o be a point wit h ± signs , a n oriente d close d curve, an d a n oriente d close d surface , respectivel y (se e Figur e 0.2) . On th e othe r hand , wha t i s a /c-for m o n a manifold ? Whe n k = 0 , i t

FIGURE 0.2 . Cycle s

is simply a function . Therefor e i t take s a valu e o n an y 0-dimensiona l cycle. Fo r genera l k > 0 , i t ma y b e sai d tha t a /c-for m i s a kin d o f function tha t ha s a value on any ordered ^-direction s (tha t is , tangent vectors) a t eac h poin t o f a manifold . Therefor e w e ca n integrat e i t over an y /c-dimensiona l cycle , an d w e obtai n a certai n quantity . W e can repea t th e theore m o f de Rham b y saying tha t w e can obtai n th e (co)homology groups , wit h coefficient s fro m R , o f an y differentiabl e manifold completel y b y suc h a n operatio n o f integratin g differentia l

•• cxO

OUTLINE AND GOAL OF THE THEORY XXll l

forms o n cycles . A s above , sinc e differentia l form s ar e something lik e functions define d o n an y ordered direction s a t eac h point, i t migh t b e easy t o understan d tha t the y ca n describ e variou s geometri c struc -tures o n manifolds . I n th e cas e o f surfaces , althoug h th e curvature , which tell s ho w muc h i t i s curved , i s a functio n o n th e surface , i t will b e mor e natura l t o conside r i t a s a differentia l for m Kda o f de -gree 2 - tha t is , a combinatio n o f th e curvatur e K an d da whic h i s called th e area l element . I t i s thi s 2-for m tha t ca n b e generalized t o the case s o f highe r dimensiona l Riemannia n manifolds . I t i s calle d the Riemannia n curvatur e form , an d i t expresse s ho w a manifol d i s curved explicitly .

Going back a little bit, we have the tangent spac e and the tangen t bundle, whic h provid e th e mos t importan t too l t o analyz e th e struc -ture o f manifolds. Th e collectio n o f al l the tangen t vector s a t a poin t is th e tangen t space , whic h give s th e firs t approximatio n describin g the state of neighborhoods of that point , and the collection of all these tangent space s over the whole manifold i s the tangent bundle . There -fore, w e ca n sa y tha t th e tangen t bundl e i s a spac e mad e o f vecto r spaces, which ar e flat spaces , over each point o f a manifold . Th e wa y these space s ar e connecte d t o eac h othe r i s controlle d b y th e grou p of al l th e automorphism s o f th e vecto r space , whic h i s a Li e grou p called th e genera l linea r group . Generalizin g thi s idea , we obtain th e notion o f fiber bundles , whic h i s motivated mainl y b y the grea t wor k of E . Carta n i n th e firs t hal f o f th e 20t h century . Briefl y speaking , a fibe r bundl e i s a manifol d obtaine d b y tyin g togethe r a famil y o f manifolds, calle d fibers , whic h stand systematicall y ove r each point of another manifol d (se e Figure 0.3) . I n fiber bundles , the group (called the structur e group ) tha t control s connection s betwee n fiber s i s a n infinite-dimensional grou p i n general , bu t th e case s where i t become s

FIGURE 0.3 . Fibe r bundl e

XXIV OUTLINE AND GOAL OF THE THEORY

a Li e group ar e especially important . The n ther e aros e a n importan t method fo r studyin g th e structur e o f differentiabl e manifolds , an d that i s to conside r variou s fibe r bundle s ove r a give n manifol d wit h various Li e groups a s thei r structur e groups , an d t o investigat e the m from a synthetic poin t o f view .

Now, how many fiber bundles, with a given Lie group as the struc-ture group , ar e there on a manifold? Thi s i s a fundamenta l question , and i t i s the theor y o f characteristic classe s tha t answer s it . Roughl y speaking, characteristi c classe s ar e a certai n descriptio n o f ho w fibe r bundles are twisted over a manifold i n terms of its cohomology groups . The characteristi c classe s calle d Cher n classe s o r Pontrjagi n classe s are typica l examples . Ther e ar e variou s approache s t o thi s theory ; among the m th e Chern-Wei l theor y i s important . Ther e i s a genera l method, wher e we give a relation betwee n th e fiber s o f a fiber bundl e in term s o f a certai n differentia l for m o f degre e 1 , called th e connec -tion, an d the n differentiat e i t t o obtai n a quantit y calle d th e curva -ture, which describes how the fiber bundl e i s curved. Th e Chern-Wei l theory give s a beautifu l framewor k fo r researc h b y systematically ap -plying this method t o fiber bundles with arbitrar y Li e groups a s thei r structure groups .

We cannot overestimat e th e importan t role s which Cher n classe s and Pontrjagi n classe s playe d i n classifyin g an d analyzin g th e struc -ture o f differentiate manifolds . Fo r example , w e hav e characteristi c numbers tha t ar e obtaine d b y integratin g polynomial s i n the m ove r manifolds, whic h ar e generalization s o f th e Eule r number . The y ca n express globa l structure s o f manifold s i n term s o f number s quit e ex -plicitly.

In moder n geometry , th e importanc e o f characteristi c classe s i s still increasing. Moreover , they are going to play a deeper role through detailed analysi s o f differentia l forms , rathe r tha n merel y cohomol -ogy classes , whic h expres s th e abov e classes . Fo r example , i n th e 21st century , ther e wil l be grand attempt s t o generaliz e th e theory o f harmonic integrals , whic h describe s th e relationshi p betwee n th e d e Rham cohomolog y an d Riemannia n metrics , i n wider frameworks . I n these ne w developments , i t i s no t to o muc h t o sa y that , s o to speak , differential form s pla y th e rol e o f wate r an d ai r fo r life .

Solutions t o Exercise s

Chapter 1 . 1.1 Th e function f m(z) ca n be written as fm(z) — g m{^)+^hrn(z),

where g m(z)1 h m(z) ar e th e rea l par t an d th e imaginar y par t o f i t respectively. O n th e othe r hand , sinc e

^(x + iy) m = m{x + iy) m~\ ^{x + iy)m = im(x + iy) m~\

we hav e

—gm = m R e f z ^ 1 ) , — gm = - m l m ^ " 1 ) .

Similarly fo r th e partia l derivative s o f h m. Therefor e th e require d Jacobian matri x i s

'mRe{zm-1) -mlm{z m-1) Kmlm{zm-1) mRe(z m-1)

1.2 Th e set M(2 ; R) of all real matrices of order 2 can be naturall y identified wit h R 4. The n 0(2 ) i s define d b y th e equatio n *AA = E (A e M(2;R)) . Her e E i s the identit y matrix . I f we let

a b vc d

the equatio n become s

a2 + b 2 - 1 - 0 , ac + bd = 0 , c 2 + d 2 - 1 = 0 .

That is , 0(2 ) i s define d b y th e abov e 3 equations i n R 4. Then , th e corresponding Jacobia n matri x i s

(2a 2b 0 0 ' I c d a b \0 0 2c 2d J

If we compute it s fou r minor s o f orde r 3 , they ar e

4a(ad-bc), Ab(ad-bc), 4c(ad-bc), 4d(ad-bc).

299

300 SOLUTIONS T O EXERCISE S

Since thes e d o no t vanis h simultaneousl y o n 0(2) , th e ran k o f th e Jacobian matri x o n 0(2) i s constantly 3 . Therefor e b y Example 1.13 , we see that 0(2 ) i s a 1-dimensiona l C°° manifold . Mor e precisely , we can prov e tha t 0(2 ) i s the disjoin t unio n o f tw o circle s S 1.

1.3 CP 1 i s obtained fro m tw o copie s o f C b y gluein g eac h subse t C — {0} , where w e remove th e origin , b y th e correspondenc e z i— > -1 . Now w e define tw o map s f± : C — > S 2 C R 3 b y

/±(z) = Irrw TTRP ' ±TTI^J (with th e doubl e sign s the sam e order) . The n i t i s easy t o verif y tha t f+(z) — /_ ( \ ) fo r a n arbitrar y non-zer o comple x numbe r z £ C . Therefore, a C° ° ma p fro m C P 1 t o S 2 i s obtaine d fro m this . W e leave i t to the reade r t o verify tha t thi s ma p i s in fac t on e to on e an d onto an d it s invers e ma p i s also o f clas s C°° .

1.4 A n arbitrar y elemen t o f SO(3) whic h i s no t th e identit y i s a rotatio n aroun d a lin e throug h th e origi n i n R 3 b y a n appropriat e angle, a s i s well known . Usin g thi s fact , w e try t o assig n a n elemen t in SO '(3) t o a n arbitrar y elemen t i n S s tha t i s a uni t 4-dimensiona l vector (a,6,c , d). I f d = ±1 , w e le t i t correspon d t o th e identit y o f SO(3). I f d ^ ±1 , sinc e (a,b,c) i s a non-zer o vecto r o f R 3, a lin e o f R3 throug h th e origi n i n th e directio n o f i t i s determined . Le t th e rotation angl e be n i f d — 0 , and a s d approaches ± 1 , let i t approac h 0 or 27r . I f we specify a n orientatio n o f R 3, th e rotatio n angl e ca n b e determined fo r exampl e by the right-han d system . I n thi s way, a ma p from 5 3 t o SO (3) wil l b e determined , an d furthermor e i t i s eas y t o see tha t ±(a , 6, c, d) ar e mapped t o th e sam e elemen t b y thi s map , s o that w e would finally obtai n a ma p R P 3 — • 50(3) . Her e we used th e fact tha t R P 3 i s obtained b y identifyin g ever y pai r ±(a , b, c, d) i n S 3. To verif y tha t thi s conjectur e i s actuall y right , w e argu e a s follows . First o f all , conside r th e cas e o f (a , 0,0, d) (tha t is , th e cas e wher e i t can b e written (sin#,0,0,cos 6)). I n thi s cas e w e le t i t rotat e aroun d the x-axi s b y th e angl e 20. The n i t goe s well , because , i f a = 0,1 , 28 = 0,7r . Th e correspondin g matri x i s

1 0 0 0 d 2 - a 2 -2ad 0 2ad d 2 - a 2

by a specia l cas e of the additio n formul a i n trigonometry . I f we argue in th e sam e wa y fo r th e case s o f a = b = 0 an d a = c = 0 an d compare th e correspondin g matrices , w e se e finally tha t a genera l

SOLUTIONS T O EXERCISE S 30 1

2 _ b 2 _ c 2 + d 2

2ab + 2c d 2ac - 2bd

2ab - led -a2 + b 2 - c 2

2ad + 2bc

element (a , 6, c, d) shoul d correspon d t o th e matri x

2ac + 2bd + d 2 -2a d - f 26c

_ a 2 _ 6 2 + c 2 + ^

belonging t o SO(3). W e leave i t t o th e reade r t o chec k tha t th e ma p thus obtaine d become s a difFeomorphism .

1.5 I f w e define a ma p / : M - > M x N b y f(p) = (p, f(p)) (p e M), obviousl y Tf = Im/ . Le t ir : M x N -^ M be the projectio n to th e first factor . The n w e hav e n o f — idM- Therefor e th e differ -ential o f / i s a n injectio n an d w e se e tha t / become s a n immersion . Furthermore, i t shoul d b e eas y t o chec k tha t / i s an embedding .

1.6 W e us e th e fac t tha t eac h componen t o f th e ma p L i s ex -pressed b y a linea r functio n i n th e coordinate s xi , • • • ,x m o f M m.

1.7 I f one uses the loca l expression (1.10 ) of the bracket, the proof of (i) , (ii ) ma y b e easy . Th e proo f o f (iv ) i s not s o difficult . Her e w e prove only the Jacob i identit y (iii) . B y the linearity of (i) , it i s enough to prov e th e cas e wher e th e loca l expression s o f X , F, Z ar e

X — /—— , Y — g——, Z = h- — OX i OX j OXk

respectively. The n w e hav e

l[X,Y],Z]

Here, fo r example , g? stands fo r —— g. I f we sum u p al l those expres -

sions, eac h o f whic h i s obtained b y applyin g a cycli c permutation o n

/ , <7 , h an d i, j , fc, we see tha t th e give n formul a become s zero . 1.8 Le t x\ , • • • ,xw an d y\ , • • • , yn b e coordinate functions aroun d

the point s p an d f(p) respectively . B y th e linearit y o f th e problem ,

we may assum e tha t v = — —. The n w e have OXi

t ( \u x^ dyj dh

'•<">*-§£;<*• On th e othe r hand , b y th e formul a fo r th e differentia l o f a composi -

tion, w e see that —— (h o /) i s also equal t o the right-han d sid e of th e

above formula , s o tha t th e give n formul a holds .

30 2 SOLUTIONS T O EXERCISE S

1.9 Le t X\ , • • • , xn an d y\ , • • • , yn b e two positive local coordinat e / dyi \

systems define d nea r th e poin t p G M. The n th e Jacobia n det ( — — J

is positive . Her e i f p € <9M , then x n = y n = 0 , s o tha t th e Jacobia n matrix i s

/ §m. . . . d vi <htL \ I dx\ r)nr , fir... \

dyi dxT)-\

dyn-i

dyi dxn

dyn_-i_ dx\ dxjt-i dx

V o •• • o fej Since obviousl y —— - > 0 . w e hav e det f — — J > 0 . There -

a t ' \OXj / l<i.j<n-l fore, al l such x\, • • • ,x„_i giv e a n orientatio n o n dM.

1.10 Firs t o f all , w e shal l se e tha t RP n i s a manifol d obtaine d by identifying eac h poin t p G Sn wit h it s antipoda l poin t — p in a n n -dimensional spher e S n. Next , i f we define / : 5" —• » S v b y f(p) = —p , we se e tha t thi s i s a n orientatio n reversin g o r preservin g diffeomor -phism accordin g t o n bein g eve n o r od d respectively . Thi s i s becaus e / ca n b e extende d t o a diffeomorphis m / o f th e whol e M n+1 b y th e same formula , an d sinc e / * ( —— ) = — ——, / preserve s th e orienta -

\oxj/ axi tion i f n i s odd , an d reverse s th e orientatio n i f i t i s even . O n th e other hand . / obviousl y map s th e outwar d norma l vecto r o f S n t o the outwar d norma l vector . Fro m thi s fact , th e abov e propert y o f / follows easily , an d w e leave i t t o th e reade r t o deduc e th e clai m fro m it.

Chapter 2 . 2.1 B y th e linearity , i t i s enough t o prov e th e assertio n fo r

LU — fdxi x A • • • A dxjk, 7 7 = gdx 3l A • • • A dxj t.

If we use the equation dxjAdxi = —dx l/\dxj repeatedly , (1 ) is proved. On th e othe r hand , fro m

LO Arj — fgdx tl A • • • A dxjk A dxJ} A • • • A dxJ}

we obtai n

d(uj A 77) = (dfg + fdg)dx u A • • • A dxn A dxj{ A • • • A dxj, .

(2) follow s fro m this . 2.2 Since ^*(ujAr]) — if*uo A^p* 77, it i s enough to prove the assertio n

in th e case s wher e UJ i s a functio n / an d dx,. Firs t i n th e cas e o f a

SOLUTIONS T O EXERCISE S 303

function, sinc e <£*(/ ) = / o </?, we hav e

On th e othe r hand , sinc e df = Y ^ —— dxi, w e hav e ^ ox z

and th e claim i s shown. Nex t w e let w = dxt. The n obviously dw = 0 , while w e hav e

J 3 k

and th e clai m i s shown . Her e w e use d th e fact s tha t th e partia l differential doe s no t depen d o n th e orde r o f differentiatio n an d tha t dyk f\dy 3 = -dyj Ady k.

2.3 B y th e Carta n formula ,

Lx{u Ar/ ) = (i(X)d + di(X)){w A ry).

Here, usin g th e fac t tha t d an d i(X) ar e anti-derivation s o f degre e 1 and — 1 respectively, w e decompos e th e right-han d sid e o f th e abov e formula an d arrang e th e result . The n (i ) i s proved, (ii ) follow s fro m

Lxduj = (i(X)d + di{X))(Lo = di{X)du

= d{i(X)d + di(X))w = dL xu>.

2.4 I f n — 2, a direc t computatio n show s tha t

J1 — 2dx\ A dx2 A dxj A dx±.

Also, i n th e cas e o f genera l n , a brie f consideratio n tell s u s tha t

ujn = n\dx\ A • • • A dx2n-

2.5 I t i s enoug h t o sho w tha t a n arbitrar y fc-form on N ca n b e extended t o a fc-form on M . I f w e pu t Uo = M \N, thi s i s a n ope n set b y th e assumption . Also , by the definitio n o f a submanifold, i f we let th e dimension s o f A/ , N b e m , n respectively , the n ther e exist s a family {Ui}i>\ o f ope n set s o f M suc h tha t eac h Uj is diffeomorphi c to E m , UiCiN = R " C M m, an d {U } n N},> i i s an ope n coverin g o f N. I n thi s case we may assum e furthe r tha t {£/,};> u i s a locally finit e open coverin g o f A/ . Le t {/,- } b e a partitio n o f unit y subordinat e t o

30 4 SOLUTIONS T O EXERCISE S

this ope n covering . No w w e pu t UJQ = 0 , an d fo r i > 1 we choos e u)i £ A k{Ui) suc h tha t

It i s clea r fro m th e for m o f U{ that thi s i s possible . I f w e pu t

= 5^/*2*> UJ

2= 0

this i s the require d extensio n o f UJ t o M . 2.6 I t i s enoug h t o sho w tha t i f f*u = 0 , the n UJ = 0 . B y th e

definition o f submersion , fo r a n arbitrar y poin t q € N an d a poin t p e M suc h tha t f(p) = q, the ma p / * : TPM — > T qN i s a surjection . Therefore, th e ma p / * : A*T*N - > A*T p*M induce d fro m i t i s a n injection. Sinc e f*uj(p) = 0 by the assumption , w e have u>(<j) = 0 . A s q can b e taken arbitrarily , w e conclude tha t UJ — 0 .

2.7 Sinc e ||x| | = y/x\ H h x , w e hav e

d||x|| = T1 —^(xidxi + h xn dx n ) . IFI I

Therefore, ^.2 I . . . i j.2 j ^

da; — —n l , , ,,— 10 n dxi A • • • A dxn + , , , , ndx i A • • • A dxn = 0 .

||:r||n+2 | |x| | n

2.8 B y Propositio n 2.13 ,

<£uUJ ~ UJ

(1) L* W = l i m * L _

Therefore i f ^ u ; = u ; fo r al l t, obviousl y w e hav e Lxw = 0 . Con -versely, suppose Lxw = 0 . Sinc e the problem i s local, we may assum e that (ftUJ i s expressed a s

^p\uj = ^2 ^ 7 ^ ' x )dxi! A • • • A dx ik

ii<---<ik

in a coordinat e neighborhood . Her e / = {i\,- • • ,ifc}. The n b y (1) , we hav e

Lxu = ^^ ~wr{0,x)dx il A-- Adx ik. ii<---<ik

Again b y (1) , w e see tha t Lx<pl& = (ft^x^ = 0 for a n arbitrar y s. Therefore, replacin g UJ b y (p* suj i n th e abov e discussion , w e see tha t

SOLUTIONS T O EXERCISE S 305

for a n arbitrar y / . Sinc e s wa s arbitrary , fi(t,x) doe s no t depen d o n t an d thu s i s a functio n onl y i n x , s o tha t ip^uj — cu.

2.9 B y th e definitio n o f th e pola r coordinates , w e can writ e r = V x2 -f - y2, 9 — arcta n — . By a direc t computatio n w e have

x

dr = (xdx + ydy) , d # = -^-— ~(xdy - ydx). ^/x2 + y2 x 2 + r

Observe furthe r tha t rd r Ad6 = dx Ady. 2.10 W e choos e

Bl = {-1 o) ' ^ 2 = ( i o) ' S 3 = ( o - t

as a basi s o f th e Li e algebr a o f SU(2). The n [Bi,B 2] = 2B 3, [B2,B3] = 2B U an d [£ 3 ,# i ] = 2B 2. Therefore , i f we let w uu2,W3 b e its dua l basis , th e require d Maurer-Carta n equation s ar e

dw\ — —UJ2 A CJ3, duj 2 = —c 3 A u;i, dcu 3 ~ — cui A UJ 2.

Chapter 3 . 3.1 Fo r (2) , i f w e pa y attentio n t o th e boundary , th e proo f o f

Theorem 3. 4 ca n b e use d withou t muc h change . Furthermore , i f w e refine th e discussio n ther e a little , w e se e tha t th e non-trivialit y o f Hn(M;Z) i s equivalen t t o th e orientabilit y o f M . (1 ) follow s fro m this. (3 ) i s easy .

3.2 A bounde d close d interva l [a , 6] is a 1-dimensiona l differen -t i a t e manifol d wit h boundary , an d f(x) i s a 0-for m o n it . Sinc e df ~ f fdx an d d[a, b) = {6 } - {a} , by th e Stoke s theore m

/ f'(x)dx= f f(x) = f(b)-f(a). J[a,b] Jd[a.b]

3.3 I f we recall the definition o f the integral of n-forms o n oriented n-dimensional C°° manifolds , (1 ) an d (2 ) ar e easy .

3.4 Sinc e ou AT? does not vanis h everywhere on M , M i s orientable. We specif y a n orientation . The n J M u A r\ ^ 0 . No w i f [UJ] = 0 , ther e exists an element 6 G Ak~1(M) suc h that dO — to. O n the other hand , since LJ A rj = d(0 A 77), by th e Stoke s theorem ,

/ UJ A 7] = / d(0 A ri) = 0 . JM JM

This i s a contradiction .

306 SOLUTIONS T O EXERCISE S

3.5 I t i s eas y t o se e that , i n general , i f a C°° manifol d M i s connected, H^, R(M) = R . Next , whil e an arbitrary 1-for m f(x)dx o n R i s a closed form , i f we put

F{x)= [ X f(x)dx Jo

we have dF = fdx, s o H lDR(R) = 0. I t i s clear tha t H^ R(R) = 0 for

k> 1 . 3.6 We shall prov e onl y tha t Hp R(Sl) = R. I f we define a map

J : H^jiiS 1) — > R b y I(UJ) — Jsl UJ, it i s eas y t o see tha t thi s i s a surjection . Thi s i s becaus e a n arbitrar y 1-for m UJ o n 5 1 ca n b e expressed a s UJ = fdx b y a functio n / : R — * R suc h tha t f(x + 1 ) =

f(x) fo r a n arbitrar y I E R , an d the n w e have /([a;] ) = / f(x)dx. Jo

Now le t us assume tha t /([a;] ) = 0 ; tha t is , / f(x)dx = 0. Then , i f Jo

we put

F{x) = / f(x)dx, Jo

F(x) become s a functio n o n 5 1 , becaus e clearl y F(x + 1 ) = F(x). Also, sinc e i t i s obvious tha t dF — fdx — UJ, w e hav e [UJ] = 0 , an d the clai m i s proved.

3.7 By usin g th e pola r coordinates , w e see that R 2 — {0 } is dif -feomorphic t o S 1 x R. Therefore , b y the homotop y invarianc e o f the de Rha m cohomology , w e have

and Hp R(Sl) i s determined b y th e previou s problem . Also , fo r ex -ample,

- y - — A x d y - y d x ) r + y

is a closed 1-for m on R — {0}, and we see that it s de Rham cohomolog y class i s not 0 (see Exercise 2. 9 of Chapte r 2) .

3.8 I f w e denot e th e uni t dis k i n R 3 b y D 3, the n <9D 3 = S 2. Therefore, b y the Stokes theorem ,

/ UJ — I duj = 3 dx\ A dx<i A dx,\ — 4n. Js2

JD* JD»

4 Here we used th e fac t tha t th e volume o f Z> * is -7r .

SOLUTIONS T O EXERCISE S 307

3.9 I f d — 0 , w e ca n tak e a constan t map . Assum e tha t d ^ 0 . Then w e take \d\ distinct point s pi (i = 1 , • • • , \d\) o n M , an d le t Ui b e mutually disjoin t smal l coordinat e neighborhood s aroun d them . The n we ca n construc t a G° ° ma p / : M — > S n suc h tha t i t map s eac h poin t Pi t o th e nort h pol e o f S n an d eac h U x t o th e norther n hemisphere , preserving o r reversin g orientatio n accordin g t o whethe r d i s positiv e or negativ e respectively , and , furthermore , al l th e remainin g par t o f M t o th e souther n hemisphere . Practically , w e can us e an appropriat e finite ope n coverin g o f M an d a parti t io n o f unit y subordinat e t o it . Now le t UJ be a n n-for m o n S n suc h tha t sup p UJ is i n th e norther n hemisphere an d furthermor e J Sn UJ = 1. The n obviousl y J M f*uj = d. Therefore b y Proposit io n 3.29 , th e mappin g degre e o f / i s exactl y d.

3.10 Le t ix : M — > M/G b e th e natura l projection . I f a ; i s a closed £>for m o n M / G , the n TT*UJ i s a close d k-iorm o n M invarian t under G . Th e correspondenc e UJ I— > TT*UJ induce s a natura l linea r ma p

(2) TT * : H*DR(M/G) - H* DR(Mf.

It i s enoug h t o sho w tha t thi s ma p i s i n fac t a bijection . Sinc e i t obviously i s fo r k — 0, w e assum e tha t k > 0 . Firs t w e shal l se e tha t it i s a n injection . Assum e tha t 7r*([u;] ) = 0 . The n ther e exist s a n element 7 7 G Ak~~l(M) suc h tha t TT*UJ — drf. No w w e pu t

1 ' gee

Here \G\ s tand s fo r th e orde r o f the grou p G . The n w e have drf = 7T*UJ.

On th e othe r hand , sinc e i t i s eas y t o se e tha t 77 ' i s invarian t unde r the actio n o f G , ther e exist s a n elemen t 7 7 G A k~1(M/G) suc h tha t 7r*77 = 7/ . Therefor e n*(df) - UJ) = 0 . Now , sinc e TT* : A*(M/G) - * A*{M) i s obviousl y a n injection , w e hav e u = dfj, an d s o [UJ] = 0 .

Next w e shal l se e tha t th e ma p (2 ) i s a surjection . Assum e tha t a d e Rha m cohomolog y clas s [UJ] represente d b y a close d /c-for m UJ o n M i s invarian t unde r th e actio n o f G . The n i f w e pu t

this i s als o a close d for m an d w e se e tha t [UJ 1] = [u;] . O n th e othe r hand, sinc e UJ' is invarian t unde r th e actio n o f G , a simila r argumen t as abov e implie s tha t ther e exist s a n elemen t UJ G A k(M/G) suc h tha t TT*LU = u/ . Sinc e UJ is obviousl y a close d form , w e ca n writ e 7r*([u>]) = [uj f] = [UJ], and th e proo f i s finished.

308 SOLUTIONS T O EXERCISE S

Chapter 4 . 4.1 Le t V b e a vecto r spac e an d fi t : V x V — » R (i = 1,2 )

two positive-definit e inne r products . I t i s simpl e t o se e that , fo r an y t £ [0,1] , ( 1 - t)fio + fy/j i s a positiv e inne r product . No w th e proo f should b e easy. W e can similarl y sho w tha t th e se t o f al l Riemannia n metrics i s contractible .

4.2 Th e invers e o f the correspondenc e give n i n th e proble m i s

• l + W TT

D 3 w H- > z = i £ H. 1 — w

In thi s case , we hav e

dw ( 1 — w)2

From thes e w e ge t

dz 2% 2\dw\ l l — "P i T 7 J

\dz\ _ 2\dw\

y i - M :

4.3 W e hav e

# = E ^T dx?

On othe r hand , i n th e isomorphis m T*R n = T xRn induce d b y th e

d dxi Eucliean metri c i t i s clear tha t dxi correspond s t o -• -

4.4 I f the loca l representatio n o f g i s hij relativ e t o anothe r pos -itive loca l coordinat e syste m (V ; y\,..., y n), the n w e hav e

>dyj KdXj'

det{gl3) = [det(^-)] 2det(fcy) ,

dx' dx\ A • • • A dxn — det( T-1)dyi A • • • A dy n.

°yj From thes e tw o equations w e hav e

\/det(gij)dxi A • • • A dxn = Jdet(hij)dyi A • • • A dy n.

If fo r a poin t p w e choos e yi suc h tha t det(/i ?J/) = 1 , the n w e ge t VM — dyi A • • • A dyn a t p , provin g th e firs t half . I t als o follow s tha t the volum e elemen t o f H 2 ca n b e writte n i n th e for m dxA

2 y .

4.5 B y definition , w e hav e

divXvM = (*d * UJX)VM = * d * a?x = ^ * ^ x •

SOLUTIONS T O EXERCISE S 309

Hence b y th e Stoke s theore m w e hav e

/ divX UA/ = / *v X-JM JdM

Now a t a n arbitrar y poin t p G DM w e choose a positive orthonorma l basis e\ ,..., e n i n T pM suc h tha t e\ = n , an d w e le t 6\ ,..., 0 n b e the dua l basis . I n thi s case , (X, TI)VQ M = (X , ei)02 A • • • A 6n. O n th e other hand , X = ]TV(X , e*)^ implie s cc; x = ^ ( X , e*)^ . Therefor e i f i : <9M C M i s the inclusion map , we get i*(*ujx) = (X , ei)#2A- • -A0„. It no w follow s tha t

/ *^ x = / (X,n)v dM, JdM JdM

completing th e proof . 4.6 B y definition , w e ge t

A / = {d5 + 6d)f = Sdf = - * d * df = -divgrad/ .

4.7 Consider , fo r example , Masse y product s arisin g fro m th e re -lation xy = yz o f thre e cohomolog y classes . I f a,/?, 7 ar e harmoni c forms representin g x,?/ , 2, the n b y assumptio n a A j5 and / ? A 7 ar e both harmonic forms . B y the Hodge theorem, they coincide with each other an d th e correspondin g Masse y produc t i s 0 . Th e proo f i n th e general cas e i s similar .

4.8 Denot e b y m,n , *M,*T V th e dimension s o f M,N an d th e Hodge operator s fo r M , N. Fo r cu e A l(M),r] e A J(N) w e ge t

*(TT*U; A 7^77 ) = ( - l ) ( m - ? ) j 7T 1 * (*A/a ; ) A 7r*{* Nr]).

If Sou — Srj = 0 , i t follow s tha t S(TTIUJ A TT] 5 ) = 0 . I f LJ,TJ ar e bot h harmonic forms , then , o f course , w e ge t d(7r*u > A 7r*77) = 0 . Henc e i n any case , w e hav e A(7r*u ; A n^rj) — 0. Th e secon d hal f follow s b y applying th e Hodg e theore m t o wha t w e just proved .

4.9 Ge t tw o copie s o f M an d le t DM b e th e manifol d obtaine d by pastin g the m alon g th e commo n boundar y dM. The n DM i s a n odd-dimensional close d manifold , an d \(DM) — 0 by Theore m 4.21 . On th e othe r hand , thin k o f th e naturall y induce d triangulatio n o f DM induce d b y tha t o f M. The n w e get \{DM) - 2\{M) - x(dM), and henc e x(M ) = \\{dM).

4.10 Th e matrice s representin g th e two intersection form s ar e (1 )

and {(? i)}-

310 SOLUTIONS T O EXERCISE S

Chapter 5 . 5.1 Fo r a trivializatio n (p : E\u = U x R n ove r a n ope n subse t

U o f M , le t rE\ f-HU) 3 (p,u) - > (p,^(u) ) € / ^ ( E / ) x R» . Thi s correspondence i s a trivializatio n o f f*E ove r ef~ l{U).

5.2 Le t n an d r ( n > r) b e th e dimension s o f E an d F . Choos e an ope n subse t U so that E\u an d F|c / ar e both trivial . The n w e can take a frame S\ ,..., s n o f E ove r U suc h tha t it s subfram e s\ ,..., s r

forms a fram e o f F|jy . No w th e n - r section s U 3 p »- > [sj(p)] £ Ep/Fp (i = r + 1 , . . . , n) for m a trivializatio n o f F / F ove r (7 .

5.3 I t consist s of two line bundles, one trivial lin e bundle an d on e non-trivial lin e bundle . Th e secon d i s obtaine d b y pastin g th e tw o ends o f [0,1 ] x R s o a s t o identif y x »- > —x.

5.4 Th e norma l bundl e o f S n i n R n + 1 i s obviousl y a trivia l lin e bundle. Henc e w e ca n combin e th e fact s tha t TS n © e = TR n + 1 | 5 n and TR n + 1 ar e trivial .

5.5 I t suffice s t o verif y directl y tha t J^ 7 A^V * satisfie s th e tw o conditions fo r connections .

5.6 Fo r UJ e A 1(M) an d s e T(E) w e hav e D{UJ <8> s) = du ® s -u <g > Vs. Henc e fo r X,Y e X(M) w e hav e

D(LU <g> s){X, Y) =]-{Xu(Y) - YLJ(X) - w([X, Y])}s

-±{UJ(X)VYS-U;(Y)VXS}

= \{VX(UJ(Y)S) - V Y(UJ(X)S) - u([X, Y])s}.

Since V s ca n b e writte n a s a linea r combinatio n o f element s i n th e form above , we hav e

D(Vs)(X, Y) = i { V x ( V y 5 ) - Vy(Vxs ) - V [x,Y]s}.

Therefore w e get D(Vs)(X, Y) = R(X, Y)s, completin g th e proof . 5.7 I f w e se t h(t) = ( 1 4- txi)(l + tx 2) •••( 1 + tx n), the n w e

get h(t) = 1 + tc\ + t 2(72 + • • • + t nan. O n th e othe r hand , fro m ft(logh{t)) = j^jh'(t), w e obtai n h(t)f t{\ogh{t)) = ti(t). Formall y we hav e

j{\ogh{t)) = x i ( l - txi + t 2x\ ) + • • •

+ xn(l - tx n +t 2xl ) .

SOLUTIONS T O EXERCISE S 311

Finally w e end up with

jt{\0gh(t)) = Sl -ts 2 + t2S3

and

(1 + tax + t2a2 + • • • + t nan){si - ts 2 + t2s3 )

= (J i + 2ta2 - f • • • + nt n~lan.

By comparin g th e coefficients o f t% w e get Newton's formula . 5.8 Us e a known propert y o f Pfaff polynomials , namely , the fact

that fo r two alternating matrice s X an d Y

Pf(o y)=Pf(X)Pf(Y)-

5.9 Fo r s e T(E*) an d t G T(E), ther e i s a function (s,t) o n M. We define a connection V * such tha t

If si,..., s n i s a loal frame field on E and 81,..., 0 n the correspondin g dual frame . I f we write VSJ = Yli 1^) ® sn the n th e condition abov e implies V*# * = ^ • —UJ^QK This show s the meaning of the problem .

5.10 (1 ) p2 = 9,p ? = 18 ; (2 ) c3 = 6 ,dc 2 = 24 , c? = 54.

Chapter 6 . 6.1 Ove r a n ope n subse t Ui = {[zi , . . . , zn+l};Zi ^ 0 } a triv -

ialization ca n b e give n b y th e correspondenc e h~ l(Ui) 3 (ZJ) J— • ([ZJ], Zi/\zi\) eUi x 51 . The action o f S1 o n the total spac e S 2n+1 i s given b y (ZJ) -» (^z) ^ e S 1 ) .

6.2 Le t [/ C B be a coordinate neighborhoo d an d let tp : n~ l(U) = [ / x F b e a trivializatio n ove r L r. I n thi s case , th e ma p tha t takes (p,u ) G ( / ' T r ) - ^ / -1 ^ ) ) t o ^(p,u ) = (/(p),*>(u) ) € £ / x F is a bijection . B y postulating suc h (p t o be diffeomorphisms, w e can define a C°° structure o n /*i£. Furthermore , (p give s a trivializatio n of f*E over f~ l{U).

6.3 B y definition o f a fiber bundle , th e projection 7 r : E - » £ i s clearly a submersion . B y Exercise 2. 6 in Chapte r 2 , 7r* : *4*(J3) — > ,4*(25) i s an injection .

6.4 Pic k a 1-cochai n d G C^if, Z) suc h tha t c - c s = Sd. Nex t take a section sf with s f = s at each vertex and satisfying the following condition. Fo r any 1-simplex K , there is a map from the oriented pat h

312 SOLUTIONS T O EXERCISE S

S'(K) • S(K;)- 1 int o S l, b y the same argument a s in the proof o f Lemm a 6.18. No w we determine S'(K) S O that it s degre e coincide s wit h s'(«) . In thi s case , we get c s> = c s - f Sd — c.

6.5 Fo r a n oriente d 2-dimensiona l vecto r bundl e n : E — > M, introduce a Riemannian metri c and choose a connection V compatibl e with th e metric . I f UJ = (ojj) an d ft = (Q lj) ar e th e connectio n an d curvature forms , w e can writ e

Moreover, w e hav e

Uj\ — —w\, Ctf = — ^ 2 , ^ 2 = C^2 -

Now i f P(E) denote s th e principa l GL(2; R ) bundl e associate d t o E, then V determines a connection UJ by virtue of Theoorem 6.50 . O n th e other hand , i f w e se t S(E) = {u E E;\\u\\ — 1}, then th e projectio n S(E) — > E i s a n oriente d S 1 bundle . W e ca n als o conside r S(E) a s a submanifol d o f P(E). Tha t is , fo r u € S(E) le t v! b e th e vecto r obtained b y 7r/ 2 rotation o f u i n th e positiv e direction . W e associat e the fram e [u,u f] t o u. I n thi s case , th e (2,1 ) componen t o f th e 1 -form tha t i s th e restrictio n o f UJ t o S(E), namely , th e portio n UJ\, i s a connectio n for m o f the principa l S l bundl e S(E), becaus e

/ 0 - l \ (cost - s i n A e X p * V l o J = U n * cos * J "

Now b y definitio n th e Eule r clas s o f E a s a n oriente d 2-dimen -sional vecto r bundl e i s represente d b y th e close d 2-for m ^ ^ 2 - O* 1

the othe r hand , th e Eule r clas s o f th e S 1 bundl e S(E) i s represente d by — ^fif. Thi s conclude s th e proof .

6.6 O n 5 3 = {z = {z\,Z2)\ |zi| 2 + \z 2\f = 1} » conside r a 1-for m UJ = x\dy\ — 2/1 da:i -f x2dy2 — y2dx2. Fo r eac h poin t z = {z\,z 2) G S3, define a ma p f z : S 1 - • S 3 b y f z(e

l°) = {z xel° ,z 2e

%e). A direc t computation show s tha t f*u = d# . Also , a; is invarian t b y th e actio n of S1 o n 53 (cf . Exercis e 6.1). I t follows that a ; is a connection form on the S 1 bundl e h : S's - > CP 1. Sinc e da; = 2(dx i Adyi + dx2 Ady 2), th e Euler clas s ca n b e determine d b y computin g — / c p i do; . No w w e

SOLUTIONS T O EXERCISE S 31 3

identify U = {[re l°', 1]} C C P 1 wit h C and define a section s : C -> S 3

by

s(re*") = ( , * re i g , , 1 ) .

Then s*(dcj ) = jj^pzyidrdO. No w we can finish th e proo f b y using :

— — / dou = ——- — Tc-zdrdO 27rJCP, 2 7 r i c ( l + r 2 ) 2

6.7 A*(so)(2) * i s a n exterio r algebr a £(<9 ) . Henc e W{so)(2) ^ £(0)®R[0].

6.8 First , tha t g~ ldg i s a lef t invarian t 1-for m wit h value s i n gl(n; IR) follows fro m

{g0g)~l d{g 0g) = g~ lgol9odg = g~ ldg

for an y go £ GL(n;R) . Second , fo r an y lef t invarian t vecto r field A = (a lj) e gi(n; R) th e value g~ldg{A) a t e is equal to (g~1dg)A = A, since {g~ 1dg)e = {dg*) and A = E a j ^ r -

6.9 Le t B\,B2, B3 be an arbitrary basi s in g = su(2) an d #1 , #2, #3 the dua l basis . Se t

and

Snz 3 Verify tha t 5Tc2 = c 2 b y direc t computation .

6.10 Fo r example , the pull-back fi*(d0) o f d8 by the map [i : Sv x S l — > S 1 tha t define s multiplicatio n i s a connectio n for m satisfyin g the conditio n i n th e problem .

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References

Some o f th e book s an d researc h article s liste d her e ar e original , historical, o r highl y recommended . Other s ar e quote d i n th e tex t a s useful o r convenien t reference s o n th e subjects .

[A] AUendoerfer , C.B. , The Euler number of a Riemannian man-ifold, Amer. J . Math . 6 2 (1940) , 243-24 8

[AW] AUendoerfer , C . B . an d Weil , A. , The Gauss-Bonnet theo-rem for Riemannian polyhedra, Trans . Amer . Math . Soci . 53(1943), 101-12 9

[BT] Bott , R . and Tu , W., Differential Forms in Algebraic Topology, Springer, 198 2

[C] Chern , S.S. , A simple intrinsic proof of the Gauss-Bonnet for-mula for closed Riemannian manifolds, Ann.o f Math. 45(1944) , 747-752

[Ca] Cartan , H. , Notion d'algebre differentielle: application aux groupes de Lie et aux varietes oil opere un groupe de Lie, 15-27; La transgression dans un groupe de Lie et dans un espace fibre principal, 57-71; Colloqu e d e Topologie , Bruxelles , Mas -son, Paris , 195 1

[deR] d e Rham , G. , Varietes Differentiables, Hermann , 195 5 [DFN] B.A . Dubrovin, B.A. , Fomenko , A.T., and Novikov, S.P., Mod-

ern Geometry-Methods and Applications, Par t I , The geome-try of surfaces, transformation groups, and fields, Par t II . The geometry and topology of manifolds, Par t III . Introduction to homology theory, Springer , 1984 , 1985 , 1990

[E] Ehresmann , C , Les connexions infinitesimales dans un espace fibre differentiable, Colloqu e de Topologie , Bruxelles , Masson , Paris, 195 1 (1950) , 29-5 5

[F] Fenchel , W. , On total curvatures of Riemannian manifolds I,. London Math . Soc . 6 2 (1940) , 243-24 8

315

316 REFERENCE S

[Fl] Flanders , H. , Differential Forms with Appliations to the Phys-ical Sciences, Academi c Press , 196 3

[GP] Guillemin,V . an d Polack , A. , Differential Topology, Prentic e Halll, 197 4

[H] Helgason , S., Differential Geometry, Lie Groups and Symmet-ric Spaces, Academi c Press , 197 8

[KN] Kobayashi , S . an d Nomizu , K. , Foundations of Differential Geometry, I , I I (Interscience) , Joh n Wiley , 1963 , 196 9

[M] Milnor , J.,O n the cobordism ring Q * and a complex analogue, Amer.J. Math . 82(1960) , 505-52 1

[MS] Milnor , J.W. and Stasheff, J.D. , Characteristic Classes, Prince -ton Universit y Press , 197 6

[Mu] Munkres , J . R . Elementary Differential Topology, Revised Edi -tion, Princeto n Universit y Press , 196 6

[NS] Nash , C . an d Sen , S. , Topology and Geometry for Physicists, Academic Press , 198 3

[S] Steenrod , N. , The Topology of Fiber Bundles, Princeto n Uni -versitiy Press , 195 1

[T] Thorn , R. , Quelques proprietes globales des varietes differentiables, Comm.Math.Helv. 28(1954) , 17-8 6

[W] Whitney , H. , Geometric Integration Theory, Princeto n Univer -sity Press , 195 1

[Wal] Wall , C.T.C. , Determination of the cobordism ring, Ann . o f Math. 7 2 (1960) , 292-31 1

[War] Warner , F.W. , Foundations of Differentiable Manifolds and Lie Groups, Springer , 198 3

[We] Wells , R.O. Jr. , Differential Geometry on Complex Manifolds, Prentice Hall , 1973 ; Springer, 197 9

Index

C, 2 1 C P " , 2 2 Cr function , 5 Cr map , 5 C°° atlas , 1 5 C^ diffeomorphism , 2 4 C°^ differen t iable horneomorphism ,

5 C°° differen t iable manifold , 1 5 C^ function , 5 , 2 3 C^ manifold , 1 5 C"*- map , 5 , 2 4 C 0 0 singula r k chain , 10 3 C°^ singula r /c-simplex , 10 3 C~*~ singula r cochai n complex , 10 4 C^ structure , 1 5 C^ triangulation , 10 1 C^ vecto r field, 9 DiffM, 4 3 ExptX, 4 3 G-structure, 23 4 GL(n;C) , 2 2 GL(n;]R), 2 0 H", 4 4 k cochain , 12 0 A;-form, 5 8 /-chain, 9 7 /-simplex, 9 6 n-dimensional numerica l space , 2 n-dimensional sphere , 1 7 n-dimensional torus , 1 7 n-dimensional vecto r space , 6 n-sphere, 1 7 0(n), 2 2 P n , 2 1 M,R2,IR3, 2

R", 2 , 6 MP", 2 1 SO(n), 2 3 TXW\ 6 e-neighborhood, 3

abstract simplicia l complex , 9 7 action o f a group , 5 0 adjoint operator , 15 4 admissible, 23 4 Alexander-Whitney map , 13 3 algebra, 24 , 5 7 alternating, 6 3 alternating form , 6 3 anti-derivation, 7 3 associated bundle , 23 6 atlas, 1 4 automorphism group , 5 0

base space , 17 1 basic elemen t (i n a Wei l algebra) ,

276 Betti number , 11 6 Bianchi's identity , 19 6 boundary, 4 5 boundary cycle , 9 8 boundary operator , 9 8 bracket, 3 9 bundle map , 232 , 23 5

Cartan formula . 7 4 Cartan-Eilenberg theorem , 13 8 Cech cohomology , 11 9 cell, 9 6 chain complex , 9 8 characteristic class , 198 , 23 8 characteristic number , 22 6

317

318 INDEX

Chern class , 20 6 Chern number , 22 5 Chern-Simons form , 28 7 classes C r , C ° ° , 5 classifying space , 23 9 closed form , 60 , 11 1 closed manifold , 4 6 cobordant, 22 6 coboundary, 9 9 cochain complex , 9 8 cocycle, 9 9 cocycle condition , 171 , 23 3 coherent orientation , 4 6 cohomologous, 9 9 cohomology, 9 8 commutative vecto r fields, 8 2 compact, 2 7 compatible (wit h a metric) , 19 9 complement o f a knot , 2 0 complete, 4 3 completely integrable , 8 0 complex Li e group , 2 2 complex manifold , 2 1 complex projectiv e space , 2 2 complex vecto r bundle , 17 1 complexification, 17 5 conjugacy, 29 0 conjugate bundle , 20 9 connection, 18 5

for a comple x vecto r bundle , 20 5 in a genera l bundle , 25 8 in a principa l bundle , 26 0

connection form , 185 , 26 4 contractible, 11 9 contractible ope n covering , 12 1 coordinate change , 1 5 coordinate functions , 1 2 coordinate neighborhood , 1 2 cotangent bundle , 67 , 17 7 cotangent space , 6 7 covariant derivative , 181 , 18 5 covariant exterio r differential , 19 3 covering, 2 7 covering manifold , 5 1 covering map , 5 1 curvature form , 186 , 188 , 252 , 26 4 cycle, 9 8

de Rha m cohomology , 11 1 algebra, 11 3 group, 11 2

de Rha m complex , 11 2 de Rha m theorem , 11 4

concerning th e product , 13 1 for triangulate d manifolds , 11 5

derivation, 3 8 diffeomorphism, 5 , 2 4 diffeomorphism group , 4 3 differentiable manifolds , 1 differential, 3 3 differential form , 5 8

coordinate-independent definition , 63

differential ideal , 8 7 directional derivative , 8 discrete group , 5 0 distance, 3 distribution, 80 , 25 8 divergence, 15 2 double complex , 12 3 dual bundle , 17 6 dual space , 6 3

elliptic PDE , 16 1 embedding, 3 4 Euclidean simplicia l complex , 9 6 Euclidean space , 14 7 Euler characteristic , 16 4 Euler class , 212 , 246 , 25 4 Euler form , 21 3 Euler number , 16 4 Euler-Poincare characteristic , 16 4 exact form , 60 , 11 1 existence an d uniquenes s o f th e so -

lution o f ODEs , 4 1 existence o f partition s o f unity , 2 9 exterior algebra , 58 , 6 3 exterior differentiation , 59 , 7 0 exterior powe r bundle , 17 7 exterior product , 59 , 6 9

face, 9 6 fiber, 171 , 23 2 fiber bundle , 23 2 flat connection , 28 8 flat G bundle , 28 8 frame field, 17 2

INDEX 319

free, 5 0 Frobenius theorem , 81 , 88 fundamental class , 10 3 fundamental vecto r field 11 9

Gauss-Bonnet theorem , 21 6 Gaussian plane , 2 2 general linea r group , 2 0 general position , 9 6 geodesic, 18 0 gradient, 14 8 graph, 5 5 Grassmann algebra , 6 3 Green's operator , 16 1

harmonic form , 15 5 harmonic function , 15 5 HausdorfF separatio n axiom , 1 1 HausdorfT space , 1 1 Hirzebruch signatur e theorem , 22 7 Hodge decomposition , 16 0 Hodge operato r * , 15 0 Hodge theorem , 15 9 holomorphic mapping , 2 2 holonomy homomorphism , 28 9 homeomorphism, 4 homogeneous coordinate , 2 1 homologous, 9 8 homology group , 9 7 homology theor y

of cel l complexes , 9 6 of simplicia l complexes , 9 6

homotopy invarianc e of de Rham co -homology, 11 9

homotopy type , 11 9 Hopf invariant , 13 4 Hopf inde x theorem , 25 6 Hopf lin e bundle , 17 4 Hopf map , 25 , 3 4 horizontal lift , 25 9 horizontal vector , 25 9 hyperbolic space , 14 7

immersion, 3 4 index, 25 6 induced bundle , 23 6 induced connection , 19 8 integrability condition , 8 8 integral curve , 3 9

integral manifold , 8 0 interior product , 7 3 intersection form , 16 6 intersection number , 16 5 invariant polynomia l function , 19 4 inverse functio n theorem , 5 , 6 involutive, 8 1 isomorphic bundles , 172 , 23 5 isolated singula r point , 25 5

Jacobi identity , 3 9 Jacobian, 5 Jacobian matrix , 5

knot, 2 0 Kronecker product , 9 9 Kunneth formula , 16 8

Laplace-Bertrami operator , 15 5 Laplacian, 15 5 left-hand system , 4 7 lens space , 5 3 Levi-Civita connection , 20 1 Lie algebra , 39 , 90 Lie derivative , 7 7 Lie group , 2 2 lift, 25 9 line bundle , 17 1 link wit h tw o components , 14 0 linking number , 14 1 local chart , 1 2 local coordinat e system , 1 , 1 2

positive, 4 9 locally finite , 2 7

manifold wit h boundary , 4 5 mapping degree , 13 9 Massey products , 13 6

triple, 13 6 Maurer-Cartan equation , 9 2 Maurer-Cartan form , 9 1 maximal atlas , 1 5 maximal integra l curve , 4 1 metric connection , 19 9 metric space , 3 multilinear, 6 3

nerve, 11 9 Newton's formula , 19 5 nonzero section , 17 2

320 INDEX

normal bundle , 17 4 null cobordant , 22 6

one paramete r grou p o f loca l trans -formations, 4 2

one paramete r grou p o f transforma -tions, 4 3

open covering , 2 7 open neighborhood , 3 open set , 3 open simplex , 11 9 open star , 11 9 open submanifold , 2 0 orbit, 5 0 orbit space , 5 0 ordered basis , 4 7 orientable, 46 , 48 , 21 1 orientation, 47 , 9 7 orientation preserving , 4 9 oriented manifold , 4 8 orthogonal group , 2 2

paracompact, 2 7 parallel alon g a curve , 18 3 parallel displacement , 18 3 partition o f unity , 2 9 Pfaffian, 21 2 Poincare disk , 16 7 Poincare duality , 16 3 Poincare lemma , 11 8 polar coordinates , 1 6 polyhedron, 9 7 Pontrjagin class , 20 0 Pontrjagin form , 20 1 Pontrjagin number , 22 5 primary obstruction , 25 5 principal bundle , 23 6 principal G-bundle , 23 6 product bundle , 17 1 product manifold , 1 6 projection, 17 1 proof o f the d e Rha m theorem , 12 6 properly discontinuous , 5 0 pullback, 72 , 23 6

quotient bundle , 17 4 quotient space , 5 0

real projectiv e space , 2 1

reducible, 23 5 refinement, 2 7 regular submanifold , 2 1 restriction o f a bundle , 17 3 Riemannian manifold , 14 6 Riemannian metric , 14 6

in a vecto r bundle , 17 5 right-hand system , 4 7

second coun t ability axiom , 1 2 section, 172 , 23 3 self-adjoint, 15 5 signature, 16 6 simplicial complex , 9 7 singular fc-chain, 10 0 singular fc-simplex, 10 0 singular chai n complex , 10 0 singular homolog y group , 10 0 singular homolog y theory , 9 6 singular poin t o f th e vecto r field, 4 2 special orthogona l group , 2 3 stabilizer, 5 0 standard /c-simplex , 9 9 Stiefel-Whitney class , 22 7 Stokes theorem , 10 7

on chains , 10 9 structure constant , 9 1 structure equation , 188 , 26 5 structure group , 23 4 subbundle, 17 4 submanifold, 2 0 submersion, 3 4 support, 29 , 10 6 symbol, 16 2 symplectic form , 9 3 system o f Pfaffia n equations , 8 8

tangent bundle , 17 0 tangent fram e bundle , 24 0 tangent space , 6 , 3 0 tangent vectors , 7 , 3 0 topological manifold , 1 3 topological space , 3 topologically invariant , 10 0 torsion tensor , 20 3 total Cher n class , 20 6 total differential , 5 9 total Pontrjagi n class , 20 1 total space , 171 , 23 2

transition function , 17 1 triangle inequality , 3 triangulation, 9 7 trivial bundle , 23 2 trivial connection , 18 5 trivialization, 17 1

unit spher e bundle , 25 6 unitary group , 2 3 universal coverin g manifold , 5 1 universal G-bundle , 23 9 upper hal f space , 4 4

vector bundle , 17 1 vector field, 9 , 3 7 vector space , 6 velocity vector , 8 vertical vector , 25 9 volume element , 15 1 volume form , 139 , 15 1

Weil algebra , 26 8 Weil homomorphism , 275 , 27 8 Whitney formula , 207- 8 Whitney sum , 17 6 Whitney's embeddin g theorem , 10 ,

36

zero section , 17 2

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Selected Title s i n Thi s Serie s (Continued from the front of this publication)

174 Ya-Zh e Che n an d Lan-Chen g W u , 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 U e n o , 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 . Si lvestrov , 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 . Osmolovskii , 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 . Kolmanovsk n an d L . E . Shaikhet , Contro l o f system s wit h

aftereffect, 199 6

156 V . N . Shevchenko , Qualitativ e topic s i n intege r linea r programming ,

1997

155 Yu . Safaro v an d D . Vassil iev , Th e asymptoti c distributio n o f

eigenvalues o f partia l differentia l operators , 199 7

154 V . V . Prasolo v an d A . B . Sossinsky , Knots , links , braid s an d 3-manifolds. A n introductio n t o th e ne w invariant s i n low-dimensiona l topology, 199 7

For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .