Complex Differential Geometry (AMS/IP Studies in Advanced Mathematics, 18) [UK ed.] 0821829602, 9780821829608

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AMS/IP

https://doi.org/10.1090/amsip/018

Studies in Advanced Mathematics Volume 18

Complex Differentia l Geometry Fangyang Zheng

American Mathematical Society

•

International Press

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Shing-Tung Yau , Managin g Edito r 2000 Mathematics Subject Classification. P r i m a r Secondary 53-00 , 53-02 , 53C55 .

y 5 3 01 ;

T h e a u t h o r wa s s u p p o r t e d i n p a r t b y N S F G r a n t # DMS-9703884 , NSA G r a n t MDA904-98-1 -0036 , a n d a fellowshi p fro m t h e Alfre d P . Sloa n F o u n d a t i o n .

Library o f Congres s Cataloging-in-Pubiicatio n Dat a Zheng, Fangyang , 1 962 Complex differentia l geometr y / Fangyan g Zheng . p. cm . — (AMS/I P studie s i n advance d mathematics , ISS N 1 089-328 8 ; v. 1 8 ) Includes bibliographica l reference s an d index . ISBN 0-821 8-21 63- 6 1. Comple x manifolds . 2 . Geometry , Differential . I . Title . II . Series . QA331.7.Z48 200 0 516.3'6—Kn\{0};

z

^ j^z

which i s clearl y a diffeomorphism . S o b y th e definition , S n i s a n n-dimensiona l different iable manifold . Another wa y to see the differentiabl e structur e o n S n i s by the implici t functio n theorem, whic h wil l b e discusse d i n th e nex t section . Th e theore m say s that , i f / i , . . . , / p ar e smoot h function s define d i n a n ope n subse t D C M m, suc h tha t

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1.2. T A N G E N T SPACE S AN D V E C T O R F I E L D S

5

dfiA-'-A df p =/z 0 along M = nf = 1 {/i = 0 } C D , the n (an y connecte d componen t of) M i s a differentiabl e manifol d o f dimensio n m — p. If w e conside r th e smoot h functio n f(xo, ..., x n) = x\ + • • • + x 2n — 1 define d on R n + 1 , the n th e zer o se t o f / i s just th e uni t spher e S n. Sinc e df = 2xodxo + • • • + 2x ndxn vanishe s onl y a t th e origi n (whic h i s no t i n 5 n ) , w e kno w tha t S n i s a smoot h manifol d b y th e implici t functio n theorem . We will leave it t o the reader s t o chec k that th e followin g space s ar e differentia l manifolds. (1) Th e rea l projectiv e spac e RP n , define d a s th e se t o f al l line s i n M n + 1 passing throug h th e origin . Not e tha t RP n = S n/((p), wher e (p is th e antipodal map . (2) Th e toru s T n = M n /Z n . I t i s th e produc t o f n copies o f th e circl e S 1 . (3) Th e compac t orientabl e surface s T^ g of genu s g , whic h i s th e connecte d sum o f S 2 wit h g copies o f T 2 ; an d th e compac t non-orientabl e surfac e o f genus g: whic h i s the connecte d su m S^#IRP 2 .

1.2. Tangen t Space s an d Vecto r Field s Let M n b e a n n-dimensiona l smoot h manifold , an d p G M. A smoot h ma p a : (—e , e)—> • M wit h a(0 ) = p i s calle d a smoot h curv e throug h p. Unde r a local coordinat e syste m ( x i , . . . , x n ) nea r p , a ca n b e expresse d a s a vector-value d function a(t) — (a\(t),..., a n(t)). No w i f (3 : (—5,5)— > M i s another smoot h curv e through p wit h th e expressio n (3{t) = (b\(t), ... ,b n(t)) unde r x , the n w e wil l sa y that a an d (5 have th e sam e initial velocity a t p if

K ( o ) , . . . X ( o ) ) = (&' 1 (o),---X(o)). Clearly, thi s conditio n i s independen t o f th e choic e o f th e loca l coordinates , an d i t defines a n equivalenc e relatio n o n th e se t o f al l smoot h curve s throug h p. W e will denote th e equivalenc e clas s [a] simply b y a'(0) , an d cal l i t a tangent vector a t p. Denote th e se t o f al l tangen t vector s a t p b y TM p, an d cal l i t th e tangent space of M a t p. I f w e writ e —|

p

= (l,0...,0,0) , ... , — |

p

= (0,0,...,0,l) ,

then unde r th e loca l coordinat e syste m x , th e se t TM p ca n b e identifie d wit h th e set o f al l linea r combination s o f {^f-| p, • • • , ^f-|p} - S o TM p i s a n n-dimensiona l vector space , an d a choic e o f the loca l coordinat e neighborhoo d ({/ , x) give s a basi s

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6

1. D I F F E R E N T I A B L E M A N I F O L D S AN D V E C T O R BUNDLE S

of TM P fo r eac h p G U. W e wil l cal l thi s 'movin g basis ' {^f-, . •., ^f- } a natural (tangent) frame i n U. A vector field X o n M assign s fo r eac h p G M a tangen t vecto r X p i n TM P, such tha t X p depend s smoothl y o n p. I n a coordinat e neighborhoo d (£/,#) , X ca n be expresse d a s a linea r combinatio n X — Yl7=i ^af~ w n e r e eac h hi i s a functio n in U. X depend s smoothl y o n p simpl y mean s tha t eac h hi i s a smoot h function . Again this i s obviously independen t o f the choic e of local coordinates. I n particular , each -£- i s a vecto r field i n U. OXi

Given a vecto r fiel d X an d a smoot h functio n / o n M , w e ca n le t X 'act ' o n / t o yiel d anothe r smoot h function , denote d b y X(f) o r simpl y Xf. I n a loca l coordinate neighborhoo d (U,x), i f X = X ^ ; ^ ~ > the n Xf i s define d b y X ^ a a T This i s agai n independen t o f th e choic e o f (U,x). W e wil l cal l Xf th e 'covarian t derivative' o f / i n X. Let u s denote b y V th e spac e of all smooth function s o n M , an d b y K th e spac e of al l vecto r fields o n M. Clearly , V i s a rin g an d ft i s a D-module . I t i s easy t o se e that K can b e identifie d wit h th e P-modul e o f al l linea r (ove r R ) map s X : V— • V satisfying th e Leibniz ' rule : X(fg) =

(Xf)g + fXg, Wf,geV.

The Lie bracket [X, Y] o f tw o vecto r fields X an d Y o n M i s define d b y [X,Y]f = X(Yf)-Y(Xf). Clearly, [Y,X] = -[X,Y] an d [X, [Y, Z}} + [Y, [Z, X}} + [Z, [X, Y}} = 0 for an y X , Y, Z i n K . (I n othe r words , ^ i s a (infinit e dimensional ) Lie algebra). A ver y importan t featur e abou t vecto r fields i s tha t the y generat e a loca l 1 parameter grou p o f loca l diffeomorphisms . Recal l tha t a local 1-parameter group of transformations o n M n i s a smoot h ma p fro m I x M int o M , (t,p) — i > t(p) , where / = ( — e , e ) , e > 0 , suc h tha t (f> t i s a diffeomorphis m o f M fo r eac h t G J, an d t ° 4>s wheneve r t,s,t +

s ar e al l i n I.

Given a loca l 1 -paramete r grou p o f transformation s {(j>t}tei o n M n , 0 o i s necessarily th e identit y map , an d {(f) t} induces a vecto r field X o n M b y (X/)(p) = l i m / ( ^ ( p ) | " / ( p ) ,

V/€D, Vpe

M

We wil l writ e thi s X a s 0o - Conversely , give n a vecto r field X o n M , fo r an y p G M, ther e exist s a n integral curve o f X passin g throug h p. Tha t is , a smoot h curve a : (—e, e)— > M fo r som e e > 0 such tha t a(0 ) = p an d a ; (t) = X Q ( t ), |t | < e . The existenc e an d uniquenes s o f thi s curv e ar e guarantee d b y th e existenc e an d uniqueness theore m fo r th e system s o f first orde r ordinar y differentia l equations . We wil l denot e b y a p (t), \t\ < e p th e integra l curv e o f X throug h p.

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1.2. T A N G E N T SPACE S AN D V E C T O R FIELD S

7

If w e fi x a poin t p G M, an d choos e a neighborhoo d U 3 p an d e > 0 s o tha t eq > e for an y q £ U. The n fo r an y £ G (—e, e), the ma p (j)t(q) = a q{t), qeU is a diffeomorphism fro m [ / ont o a neighborhood M , suc h tha t th e following holds : There exist s a n ope n coverin g {U a}aeA o f M , indexe d b y a se t A, an d fo r each a G i a diffeomorphis m f a : 7r-1(Z7a)— > U a x W suc h tha t wheneve r U ab : = UanUb^ 0 , the map fa of' 1 : Uah x W - » U ab x R r send s (p , v) to (p , g ab(p){v)), where g ab : C/ a6— > GL(r, R) i s a smoot h ma p whic h take s value s i n th e genera l linear grou p o f order r . In othe r words , locall y ir looks lik e th e projectio n ma p from U x l r ont o U\ and whe n patchin g up , the linear structure s o n the fibers are preserved. Not e tha t E nee d no t be homeomorphic t o the product M x R r . For p £ M , th e invers e imag e TT~ 1 (P) = R r i s calle d th e fiber o f E a t p, denoted b y E p. Th e GL(r 1 R)-valued function s g ab on ?7 a6 ar e calle d transition functions o f th e vecto r bundle . The y satisf y th e cocycl e conditio n tha t o n eac h Uabc = U anUbC\UCl g ab9bc9ca = I (i-c , it i s constantly th e identity matrix . Her e we adop t th e convention tha t g aa = I). Conversely, give n a n ope n coverin g {U a}aeA o f M n an d a GL(r, R)-valued smooth functio n g ab on eac h U ab satisfying th e abov e cocycl e condition , the n w e can us e gab to patch u p those U a x R r an d get a rank r vecto r bundle . If w e replace R r b y C s , the complex Euclidea n spac e o f complex dimensio n s , and requir e g ab to tak e value s i n the complex genera l linea r grou p GL(s,C), the n we get the definition o f a complex vecto r bundl e ove r M. 1 .5 . Le t n : E — • M n b e a ran k r vecto r bundl e ove r a differ entiate manifol d M n , an d U a n ope n subse t i n M. A section o f E ove r U i s a differentiate ma p s : U —> E suc h tha t TT o s = idjj i s the identity map. DEFINITION

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1.3. V E C T O R BUNDLE S

9

We wil l denot e b y I\Z7 , E) th e se t o f al l section s o f E ove r U. Naturally , i t form s an (infinit e dimensional ) vecto r spac e ove r M.. I n fact , i f w e denot e b y C°°(U) th e ring o f al l smoot h function s define d o n [/ , the n T(U,E) i s a C°°(Z7)-module . If { s i , . . . , s r} C r(f7 , E) i s a se t o f section s suc h t h a t { s i ( p ) , . . . , s r(p)} form s a basi s o f E p fo r ever y p G U, the n w e sa y { s i , . . . , s r} i s a frame o f th e bundl e E over U. Using th e transitio n function s {g ab}, i t i s clear t h a t a globa l sectio n s G T ( M, E) is jus t a collectio n o f smoot h R e v a l u e d function s a a o n U ai fo r eac h a G A , suc h t h a t a a — gabO'b o n U ab fo r al l a , b G A T h e usua l construction s fo r vecto r space s ca n b e carrie d ove r t o th e bundles . Let u s first recal l th e construction s i n th e vecto r spac e case . Suppose t h a t V an d W ar e tw o finite dimensiona l vecto r space s ove r a field F. (F wil l usuall y b e 1 o r C fo r ou r purposes) . Conside r th e infinit e dimensiona l vector spac e M(V,W) ove r F whic h ha s th e se t V x W a s a basis , i.e. , th e fre e vector spac e generate d b y th e pair s (v, w) fo r al l v G V an d w G W. Le t N b e th e subspace o f M(V , W) spanne d b y al l th e element s o f th e for m (kv, w) — k(v, w), (v

+ v' , w) — (v, w) — (vf, w),

(v, kw) — k(v, w), (v,

w + w') — (v, w) — (y, w f)

where v, v' G V, w, w' G W an d k G F. Th e quotien t spac e M(V , W)/N i s called th e tensor produc t o f V an d W, an d i s denote d b y V 0 W. Fo r v e V an d u > G W, thei r tensor produc t i s v®w := [(v, w)], th e imag e o f (v, w) unde r th e n a t u r a l projectio n from M t o M/N. Th e ma p ip: VxW^VW define d b y If { i > i , . . . , v n } i s a basi s o f V an d { u > i , . . . , w m } i s a basi s o f W, t h e n th e se t {vi 0 w a; I < i < n,l < a < m} become s a basi s o f V 0 W. I n particular , V 0 W has dimensio n n • ra. It i s eas y t o se e t h a t V 0 W = V F 0 V ^ (vi a th e m a p v 0 w h- » i u 0 i>) , an d (V 0 W) 0 E / = V 0 ( W 0 f7) . S o i t make s sens e t o writ e thing s lik e V 0 W 0 U. Consider th e tenso r product o f g-copie s o f V:

V®q := V 0 V 0 • • • 0 V vv

For an y q vector s v\ product b y

/

,.. ., v q i n V , defin e th e symmetric product

^1 A ^2 A • •

1

V -

1

V -

an

d alternative

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10

1. D I F F E R E N T I A B L E M A N I F O L D S AN D V E C T O R BUNDLE S

where the sum i s taken ove r all the permutations n of {1 , 2 , . . ., q} : an d e n i s the sign of the permutation (e n — 1 or — 1 when TT ca n be decompose d int o the product o f an even o r od d numbe r o f interchange s - permutation s tha t onl y mov e two positions) . Clearly, th e symmetri c (alternative ) produc t i s symmetri c (skew-symmetric ) with respec t t o an y tw o o f it s positions . Th e alternativ e produc t i s als o calle d th e wedge product o r exterior product I t i s non-trivial onl y whe n q is less than o r equa l to th e dimensio n o f V. The vecto r spac e spanne d b y th e symmetri c o r wedg e product o f an y q vectors in V i s called th e q-th symmetric o r exterior product o f V, denote d b y S qV o r A qV, respectively. For a vecto r spac e V ove r a field F , it s dua l spac e V* i s th e on e consistin g o f all th e linea r map s fro m V int o F. Finally, fo r matrice s A = (A i3) G GL(r,F) an d B = (B ap) G GL(s,F), defin e their direc t su m A © B G GL{r-f - s, F) an d tenso r produc t A® B e GL(rs, F) b y A®B=(0

J , (A®B)

B

iaij(3

=

A ijBolf3

respectively. Also , defin e th e q-th exterio r (wedge ) produc t o f A b y (AM) JJ = 2_^ eirAiljn(1) •

• • AiqJ7v(q)

where I = (ii,...,i 1 < i\ < • • • < i q < r an d J = ( j i , . . . ,j q ), 1 < j i < q), • • • < j q < r ar e multi-indices , an d th e su m i s taken ove r al l th e permutation s TT o f { 1 , . . . , q}. Similarly , th e q-th symmetri c produc t o f A i s defined b y

(SqA)u =

2 ^ ^7J) Ai^^) ' ' TV ^

' Ai ^(q)

'

where / = ( z l 5 . . . , z g), J = ( j i , . . . ,j q) wit h 1 < i\ < • • • < i q < r an d 1 < j i < • • • < jq < r - Th e sum is over all permutations 7 r of { 1, . . ., g} , an d a (I) = n\\ - - -n v\ if thos e ik take s p distinc t values , wit h r i i , . . . , n p element s i n eac h group . Tha t is , ^1 : = • • • = = l n i
M an d n f : E'— > M' ar e vecto r bundles , and g : M— > M ' a smoot h map . A smoot h ma p f \ E —> E f is calle d a bundle homomorphism (or bundle map) over the base map g, i f fo r an y p G M, / | ^ p i s a linear ma p fro m j& p into E f, y I n th e cas e M = M f an d g i s th e identit y map , / is simpl y calle d a bundle map.

1.4. Tangen t Bundle s an d Tenso r Field s

Perhaps th e mos t importan t clas s o f vecto r bundle s i n differentia l geometr y i s the clas s o f natural bundles, namely , thos e whic h ca n b e derive d fro m th e tangen t bundle throug h finitel y man y step s o f direc t sum , tenso r product , wedg e product , symmetric products , an d duals . Th e tangent bundle o f a differentiabl e manifol d M n , denote d b y TM, i s the unio n o f the tangen t space s TM P fo r al l p G M. . 7 (Tangen t Bundles) . Le t M n b e a differentiabl e n-manifold , cov ered b y a collectio n o f coordinat e neighborhood s {U a,xa = (x ai,..., x an)\ a G A}, with inde x se t A. The n {^—, . • • , ^r—} give s a fram e o f TM ove r U a. I t i s called th e natural (tangent) frame o f th e coordinat e neighborhoo d (U a,xa). I n a non-empty overla p U ab — Ua nUb, th e transitio n functio n o f TM i s give n b y th e coordinate chang e matrix : = fdx ai\ 9ab ~ \dx b3)nxn The dua l bundl e o f TM i s calle d th e cotangent bundle o f M , denote d b y TM*. EXAMPLE 1

Note tha t th e section s o f th e tangen t bundl e TM ar e exactl y th e vecto r field s in M. Mor e generally , w e wil l cal l a sectio n Y o f th e natura l bundl e {TM*)® r (TM)®S a tensor field of bidegree (r, s). Her e r an d s ar e nonnegativ e integers , called th e covariant an d contravariant degree o f F , respectively . Under th e loca l coordinat e syste m ( x i , . . . , x n), w e hav e a loca l tangen t fram e { af- ' • • *' a f - } an< ^ c °tangent fram e {dx\,... , dx n}. S o a tensor fiel d Y o f bidegre e (r, s) ca n b e writte n a s n

Y= V

Yl

1

-/Sdxll(g

> • • • 0 cb i r (g )(g

> • • • (g )

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12

1. D I F F E R E N T I A B L E M A N I F O L D S AN D V E C T O R BUNDLE S

in th e neighborhood . Th e smoot h function s Y? 1 '"?8 ar e calle d th e components o r coefficients o f Y unde r th e loca l frames . Denote b y 7 J th e P-modul e o f al l th e tenso r fields o f bidegre e (r , s) o n M. Then 7^ ° i s just N , the Li e algebr a o f al l vecto r fields o n M. A very importan t typ e o f tensor fields ar e th e differentia l forms . Fo r a positiv e integer p < n , a (differential) p-form ip on M n i s a sectio n o f th e bundl e A PTM*. Equivalently,

A

makes A*(M) int o a n algebra . I t i s calle d th e exterior algebra of M. W e hav e ip A ip = (-1 )^- 0 A if whe n cp G Ap an d ip G A q. The exterior differentiation operator , denote d b y d, whic h send s a p-form t o a (p + l)-form , i s determine d b y th e rule s tha t d i s linea r ove r R an d ^ Pi

d(fdxil A

-p

• • • A dx ip) = ^^ Q— dxj A dx h A • • • A dx ip. .7 = 1

J

UX j

It i s eas y t o verif y tha t d2 = 0 ; an

d d(cp A ip) = dip A ip + ( - 1 ) ^ A dip

if (p is a p-form . A p-for m ip is calle d a closed form, if dip — 0. W e wil l denot e by Z v th e subspac e o f A p consistin g o f al l th e close d p-forms , (p i s calle d a n exact p-form i f ther e exist s a ( p — l)-for m ip such tha t ip = dip. Sinc e d 2 = 0 , dA p~l i s always a subspac e o f Z p . For ip • i 7 i s a Lie group homomor phism, the n /* e : g—> • f ) i s a Li e algebr a homomorphism , an d / e x p G = exp ^ /* e . In particular , i f G i s a Li e subgrou p o f iJ , the n g C { ) is a Li e subalgebr a an d exp G = exp ^ | fl. W e wil l als o as k th e reader s t o verif y tha t fo r th e matri x groups , the exponentia l ma p i s just th e usua l matri x exponential . For g G G, conside r th e inne r automorphis m o-g : G -> G , c^O ) = gxg~

l

ag i s a n automorphis m o f G . It s differentia l a t e , denote d b y .Ac^ , i s a n automor phism o f g = Lie(G). Th e Li e grou p homomorphis m Ad:G^GL{g), g

^ (or g)*e

is calle d th e Adjoint representation o f G . Th e kerne l o f thi s representatio n i s exactly th e cente r Z o f G . Taking th e differentia l o f Ad a t e , w e get a Li e algebra homomorphis m ad fro m g int o th e matri x algebr a g/(g) . Agai n th e kerne l i s just th e cente r (ideal ) 3 of g. We will leav e i t a s a n interestin g exercis e fo r th e reader s t o sho w tha t adx{Y) =

[X,Y], V I , 7 G

0

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20

1. DIFFERENTIABL E MANIFOLD S AN D VECTO R BUNDLE S

EXAMPLE 1 .1 1 . Conside r th e grou p SL(2,R ) o f al l rea l 2 x 2 matrice s wit h determinant 1 . It s Li e algebr a i s 5/(2 , R) o f al l traceles s rea l 2 x 2 matrices . Fo r A G 5/(2 , R), le t u — yj\ de t A\. The n b y similarit y reductio n o f A t o th e diagona l or specia l triangula r form , i t i s easy t o sho w tha t

{

cos u l +^ A, i c o s h u J + ^ , 4 ,i I + A, i f d e t

f detA>0 f det,4< 0 4= 0

From this , i t i s not har d t o see that fo r B G SL(2, R) , i f Tr(B) > - 2 , ther e exist s a uniqu e A G 5/(2, R) suc h tha t e A = B, i f Tr(B) < - 2 , ther e i s n o suc h A, an d when Tr(B) = —2 , there ar e infinitely man y suc h A. S o in general, the exponentia l map i s neithe r injectiv e no r surjective . The Killing form o f a Li e algebra g is the symmetri c bilinea r for m o n g define d by th e trac e o f the compositio n B(X,Y) =

Tr(ad xadY), X,Y

G

g

It i s not har d t o see that fo r an y X, Y, Z G g and an y automorphis m / o f g, it hold s B(X,[Y,Z]) = B(fXJY) =

B(Y,[Z,X}) B(X,Y)

=

B(Z,[X,Y})

DEFINITION 1 .1 2 . A Li e algebr a g i s calle d semisimple i f it s Killin g for m i s non-degenerate. A non-abelian Li e algebra g is called simple i f 0 and g are the onl y ideals i n g .

Denote b y N = {X G g | B(X,Y) = 0, V Y G g} th e nul l spac e o f B. N i s always a n idea l i n g . S o a simpl e Li e algebr a i s alway s semisimple . I n th e exercise , we shall se e tha t a semisimpl e Li e algebr a i s alway s th e direc t su m o f simpl e ones .

Appendix: Topology , Homotop y an d Coverin g Space s

We includ e thi s appendi x her e fo r th e convenienc e o f thos e reader s wh o hav e not studie d genera l topolog y before . A topological space is a non-empty se t X equippe d wit h a collection T o f subsets in X, suc h that T contain s the elements X, 0 (th e empty set), an d T i s closed unde r the finite intersectio n an d arbitrar y unio n operations . Th e set s i n T ar e calle d th e open subsets. Th e complemen t o f a n ope n subse t i s calle d a closed subset Of cours e thi s i s a ver y broa d concept , an d include s man y strang e specimens , such a s T = {X, (/>} , o r T consist s o f al l th e subset s o f X. Fo r man y purposes ,

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A P P E N D I X : T O P O L O G Y , H O M O T O P Y AN D C O V E R I N G SPACE S 2 1

the firs t topolog y i s to o small , whil e th e secon d on e (whic h i s calle d th e discret e topology o n X) i s often to o large . S o we will restrict ourselve s t o mor e 'reasonable ' topological spaces . A topological spac e X i s called Hausdorff, if for an y tw o distinc t point s x 1 y i n X, ther e exis t ope n subset s U 3 x an d V 3 y suc h tha t U C\ V = 0 , denote b y B r(x) th e se t o f al l point s i n R n whos e (Euclidean ) distanc e fro m x i s less tha n r . I t i s a calle d th e (open ) bal l o f radiu s r centere d x. Le t B b e th e collection o f al l B r(x) wit h rationa l r an d rationa l x (i.e. , eac h componen t o f x i s a rationa l number) . Th e topolog y generate d b y B i s called th e Euclidea n topology . Clearly i t i s Hausdorff, an d ha s a countable se t B a s it s base . Ther e ar e a fe w mor e terminologies tha t ar e frequentl y used . (1) A map / : X— > Y betwee n two topological spaces is said to be continuous, if f~~ l (U) i s open for any open subset U of Y. / i s calle d a homeomorphism if i t i s a bijectio n an d bot h / an d f~ l ar e continuous . (2) A spac e X i s sai d t o b e connected, i f i t canno t b e writte n a s th e disjoin t union o f two non-empty ope n subsets , i.e. , i f X an d • X wit h /(0) = / ( l ) = x. Le t u s denot e i t b y / . I f ~g i s anothe r suc h loop , w e ca n compos e them t o ge t a ne w one , denote d b y / * ~g, i n th e followin g wa y

f f(2t), 0 • 7Ti(Y , f(x)). A path-connecte d spac e wit h TTI(X ) = 1 connected) space .

i s calle d a simply-connected (o r 1 -

Let X , Y b e path-connecte d spaces . A ma p / : X — » Y i s calle d a covering map, i n whic h cas e X i s called a covering space of Y , i f / i s continuous, surjective , and fo r an y y £Y, ther e exist s a connecte d ope n subse t y G U C Y, suc h tha t eac h connected componen t o f f~ l(U) i s homeomorphi c t o U unde r / . The basi c featur e abou t coverin g space s i s the followin g liftin g property , whos e proof i s lef t t o th e reader s a s a n exercise . Lifting Property : Suppose f : X — > Y is a covering map. If Z is pathconnected and locally path-connected, and g : Z — > Y is a continuous map with g(z0) = f(x 0) and 0*(7Ti(Z,z o)) C f*(iri(X,xo)), then for any x x G f~ 1 {f[x[))), there exists a unique continuous map g : Z— > X such that g(zo) = x\ and fog = g. Recall that a topological spac e Z i s called locally path-connected if , fo r an y ope n set U C Z an d an y z £ U, ther e exist s a n ope n se t V wit h z G V C [/ , suc h tha t any tw o point s i n V ca n b e connecte d b y a pat h i n U. In particular , an y pat h connecte d an d locall y pat h connecte d spac e Y admit s a uniqu e simply-connecte d coverin g space . I t i s called th e universal covering space of y , ofte n denote d b y Y.

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23

EXERCISES

Exercises (1) Prov e that , i f X i s Hausdorff, the n an y compac t subse t o f X i s closed . (2) I f / i s a bijective , continuou s ma p fro m a compac t spac e X ont o a Hausdorf f space Y. Sho w tha t f~ l i s continuous . (3) {Metric Spaces). A metric (or distance) space is a n non-empt y se t X togethe r with a function d : I x I ^ R satisfyin g th e propert y tha t fo r an y x,y,z G X, d(x,y) = d(y,x) (symmetry) , d(x,y) > 0 an d = 0 if f x = y (positivity) , an d d(x,z) < d(x,y) + d(y,z) (triangl e inequality) . Suc h a functio n d i s calle d a distance o n X. For x G X an d r > 0 , denote b y B r(x) th e se t o f all y G X wit h d(x , y) < r . It i s calle d a n (open ) bal l (o f radiu s r centere d a t x). Th e topolog y generate d by the se t o f all open balls is called th e topology o f d, denote d b y 7^. Sho w tha t Td is alway s Hausdorff , an d i t ha s a countabl e bas e i f an d onl y i f X contain s a countable subse t A C X whos e distanc e t o X i s 0 (i.e. , give n an y x G X an d any e > 0 , ther e exist s a G A wit h d(a, x) < e) . (4) Sho w tha t th e Euclidea n distanc e functio n d on R n define d b y d(s,y) = £ > - j / i )

2

)*

i=l

satisfies th e triangl e inequality : d(x, z) < d{x, y) + d(y, z) fo r an y thre e point s in R n (s o i t i s a distance , sinc e th e othe r tw o condition s ar e clearl y satisfied) . (5) Sho w that a subset i n R n i s compact i f and onl y i f it i s closed an d bounded (i.e., it i s containe d i n a larg e ball) . (6) Sho w tha t th e subspac e X = {(0,0) } U {(x,y) : connected bu t no t path-connected .

x > 0, y = si n ±} o f R 2 i s

(7) Le t X b e a path-connecte d space . Le t 7(X ) b e th e se t o f homotopy equivalen t classes o f loops i n X (ignorin g bas e points) . I f we fix a base poin t x G X, ther e is a natura l ma p 7Ti(X,x) —> j(X) givin g b y ignorin g th e bas e point . Sho w that thi s ma p i s a epimorphis m (i.e. , surjectiv e homomorphism) . Whe n i s thi s map a monomorphis m (i.e. , injective) ? (8) Le t X b e a topologica l space , ~ a n equivalenc e relatio n i n X , an d Y — X/ ~ the quotien t set . Denot e b y IT : X — • Y th e projectio n map . Th e quotient topology on Y i s the larges t topolog y suc h tha t TT i s continuous. I n othe r words , under thi s topology , U C Y i s ope n i f an d onl y i f 7r~ 1 U i s open . Giv e a n example o f X an d ~ s o that X i s Hausdorf f bu t Y i s not . (9) Le t X b e a topologica l spac e an d G a grou p actin g o n X (i.e. , G i s a subgrou p of th e homeomorphis m grou p o f X). Denot e b y Y = X/G th e quotien t spac e (i.e., th e orbits) . Th e actio n o f G i s sai d t o b e free i f an y e ^ g G G ha s n o fixed point s i n X. G i s sai d t o ac t properly discontinuously i f an y p e X ha s a neighborhoo d U 3 p suc h tha t th e subgrou p {g G G \ U D gU ^ 0 } i s finite .

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1 24

. D I F F E R E N T I A B L E M A N I F O L D S AN D V E C T O R BUNDLE S

Prove t h a t , i f G act s properl y discontinuousl y o n a Hausdorf f spac e X , the n the quotien t spac e i s als o Hausdorff . I f G i s bot h properl y discontinuou s an d free (an d X i s p a t h connecte d an d locall y p a t h connected) , the n TT : X—- > Y i s a coverin g map . (10) Le t X b e th e quotien t spac e (equippe d wit h th e quotien t topology ) o f MP 3 b y the relatio n ( # i , £ 2 , ^ 3 , £4 ) — (—#2? #i 5 — x4 , #3) . C o m p u t e 7Ti(X) . (11) I f G an d i 7 ar e tw o finite group s actin g freel y o n a spac e X , an d i f th e action s commute (i.e. , g o h — h o g fo r an y g £ G an d h 0 , X n i s obtaine d fro m X n _ i b y attachin g a collectio n o f n-cell s (i.e., X n \ X n _ i = Ue A i s th e disjoin t unio n o f a collectio n o f ope n subset s e\, an d ther e exis t fo r eac h A a continuou s m a p f\ : B n -^ ~e~\ whic h map s the uni t bal l B n — ^i(O ) C M n homeomorphicall y ont o e\. Eac h e\ i s called a n (open ) n-cel l i n X n). (c) X an d eac h X n hav e th e weak topology, namely , a subse t A o f X (o r X is close d i f an d onl y i f A D e i s close d fo r eac h cel l e .

n)

Xn i s calle d th e n-skeleton o f X . Not e t h a t th e conditio n (c ) i s relevan t only whe n ther e ar e infinitel y man y cell s i n X . Also , fo r som e n , ther e nee d not b e an y n-cell s (i.e. , X n = X n _ i ) . Th e abov e CW-comple x structur e i s sai d to b e n-dimensional, i f X = X n ^ X n _ i . I t i s calle d a finite CW-complex i f there ar e onl y finitely man y cell s i n X . Show t h a t fo r an y CW-comple x X , i t hold s TTI(X ) = 7Ti(X2) . Also , thi s group ca n b e presente d i n th e wa y t h a t eac h 1 -cel l represent s a generato r an d each 2-cel l represent s a relatio n (i n particular , i f X 2 i s a finite CW-complex , then 7Ti(X ) i s a finitely presente d group) .

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EXERCISES

25

Prove tha t an y finitely presente d grou p i s th e fundamenta l grou p o f a 2 dimensional finite CW-complex . (16) Notic e tha t th e fundamenta l grou p o f a topologica l spac e i s jus t th e (fixe d base) homotop y equivalen t classe s o f continuou s map s fro m S 1 int o th e spac e (sending 1 to the base point). I f we replace S 1 b y the spher e S n, w e get a grou p 7r n (X,x) calle d th e n-th homotopy group of X. (a) Verif y th e detail s an d sho w that , fo r n > 2 , 7r n(X) i s alway s a n abelia n group. (b) Le t / : X—> • Y b e a covering map . Sho w tha t fo r eac h n > 2 , / induce s a n isomorphism / * : 7rn(X) = ir n(Y). (17) Sho w tha t 7r n (X), H q(X,R) an d H q(X,R) ar e al l homotopy invariant, i.e. , any homotop y equivalenc e / : X — > Y (meanin g tha t / i s continuous an d ther e exists a continuou s ma p g : Y— » X suc h tha t / o g i s homotopi c t o idy, g o / is homotopi c t o idx) induce s isomorphism s o n th e correspondin g group s o f X and Y . (18) (Simplicial Complexes). Recal l tha t a n n-simple x a i s th e conve x hul l o f n + 1 points i n som e Euclidea n spac e R m suc h tha t th e dimensio n o f a i s n (I n othe r words, thes e point s ar e i n genera l positions) . Thos e n + 1 points ar e calle d th e vertices o f a. Fo r k < n, an y k + 1 of thes e vertice s for m a /c-simplex , whic h i s called a (/c-dimensional ) face o f a. A (finite ) simplicial complex i s a finite collectio n K o f simplexe s i n som e R m , suc h tha t th e face s o f an y simple x i n K ar e als o i n K, an d fo r an y tw o simplexes a an d r i n K wit h a f l r ^ 0 , th e se t a H r i s a commo n fac e o f a and r . The underlyin g topologica l spac e o f K i s often denote d b y \K\. W e some times sa y that K i s a simplicial comple x structur e o n it s underlyin g topologica l space. Th e larges t dimensio n o f al l simplexe s i n K i s calle d th e dimension o f K. Not e tha t a simplicia l comple x i s a finite CW-complex , bu t th e convers e i s not true . If we allow K t o b e a n infinit e bu t locally finite collection , i.e. , an y simple x in K onl y intersect s wit h finitely man y othe r simplexe s i n K, the n w e ge t th e concept o f infinite simplicial complex. Th e differenc e is , i n thi s cas e w e ma y not b e abl e t o pu t al l o f th e simplexe s i n on e Euclidea n space . I n fact , th e dimension o f th e simplexe s i n K coul d eve n b e unbounded . (19) Le t X b e a simplicia l complex . Denot e b y S q(K) C S q(\K\) th e subspac e whic h is the fre e abelia n grou p generate d b y al l th e g-simplexe s o f K. The n S*(K) C S*(\K\) i s a subcomplex . Sho w tha t th e inclusio n ma p induce s isomorphism s on th e homolog y groups . Thi s give s a combinatoria l wa y o f computin g th e homology o r cohomolog y group s o f th e underlyin g topologica l spac e o f K. I n particular, th e Eule r numbe r o f \K\ i s equal t o XX -^)qnq-> wit h n q th e numbe r of g-simplexe s i n K.

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1 26

. D I F F E R E N T I A B L E M A N I F O L D S AN D V E C T O R BUNDLE S

(20) Suppos e X — YUf e n i s obtained b y attaching a n n-cel l to Y (vi a the boundar y map / : S 71 '1 - • Y). Sho w tha t H q(X) = H q{Y) whe n q / n o r n - 1 , an d there exist s a n exac t sequenc e 0 -> H n(Y) - tf

n(X)

- > Z ^ Hn-^Y) -+

ff n-i(X) - 0

where / * i s th e induce d ma p o f / o n th e (n — l)-t h homolog y group , an d w e wrote H q{X) fo r H q{X,Z). (21) Sho w tha t th e e(X x Y) = e(X) • e(Y), provide d tha t al l term s exist . Wha t about th e bett i numbers ? (22) Prov e tha t fo r an y give n continuou s ma p / : S 2— > R2 , ther e alway s exist s x e S 2 suc h tha t f(—x) = f(x). (23) Triangulat e th e Mobiu s band , an d us e i t t o comput e th e integra l homolog y group H 2. (24) Sho w that fo r a covering ma p / : X — > Y o f finite degre e m (i.e. , the se t f~ has m elements) , th e Eule r number s satisf y e(X) = m • e(Y).

l

{y)

(25) Giv e a simplicia l comple x structur e o n (i.e. , triangulate ) T, g. (26) Conside r th e connecte d su m M#N o f tw o n-dimensiona l manifolds . Fin d th e fundamental grou p an d th e homolog y group s o f M#N i n term s o f thos e o f M and N. Prov e tha t r(M±m-l 1w

}

e

( M ) + e (W)> i ~ \ e(M ) + e(N) - 2 , i

f n i s odd , f n i s even .

In particular , e(E g) = 2 — 2g. (27) Construc t a C°° diffeomorphis m / fro m th e bal l i?3 r(0) C W 1 ont o itself , suc h that / i s the identit y ma p i n th e shel l J?3 r(0) \ #2r(0) 5 an d /(0 ) = (r , 0 , . . ., 0) . Use this ma p t o sho w that th e grou p o f all diffeomorphisms (homeomorphisms ) of a differentiabl e (topological ) manifol d act s transitivel y o n th e manifold . (28) Le t E — > M an d F — > N b e vecto r bundle s ove r differentiabl e manifolds , an d / : E— > F b e a bundl e ma p ove r th e bas e ma p h : M— > N. Sho w tha t i f / i s injective, the n s o i s h. (29) Le t X an d Y b e vector fields on M n. Suppos e (j) t i s the local 1 -paramete r grou p of loca l transformation s generate d b y X. Sho w tha t fo r an y p E M , [ X , y L = li m

y p - ((4> t)*Y)f

(30) Verif y th e lon g formul a fo r drj where rj is a p-form . (31) Le t M n b e a compac t smoot h manifol d an d X b e a vecto r field o n M. Sho w that X i s complete , i.e. , i t generate s a loca l 1 -paramete r grou p o f globa l dif feomorphisms o n M. I f X i s nowher e zero , sho w tha t i t generate s a globa l

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EXERCISES

27

1-parameter grou p o f diffeomorphism s o n M (i.e. , {cj) t} where t i s define d i n the entir e R) . (32) Le t G b e a Li e group , an d f ) b e a Li e subalgebr a o f Lie(G) (i.e. , i t i s a linea r subspace an d i s close d unde r th e Li e bracket) . Sho w tha t ther e exist s a Li e subgroup H o f G suc h tha t Lie(H) = f) . (33) Sho w tha t ad xY =

[X,Y].

(34) Sho w tha t th e exponentia l ma p o n GL(n, C ) i s surjective . (35) Prov e tha t a close d subgrou p o f a Li e grou p i s a Li e group . (36) Prov e tha t ther e ar e onl y tw o (u p t o isomorphism ) Li e algebra s o f dimensio n 2: th e abelia n on e an d th e on e define d b y g = R{X , F }, [X , Y] ~ X. Classif y all thre e dimensiona l Li e algebras . (37) Sho w tha t 5/(2 , R) i s simple, an d i t ha s a basi s {X,Y, Z} satisfyin g [X, Y] = Z, [X , Z] = - 2 X , [y , Z] = 2Y. (38) Le t a b e a n idea l i n a semisimpl e Li e algebr a g . Denot e b y a 1 - the orthogona l complement o f a i n g with respec t t o th e Killin g for m B. Sho w tha t a 1 - is als o an ideal , bot h a and a 1 - are semisimpl e Li e algebras, an d g = a © a-1 i s a direc t sum. S o any semisimpl e Li e algebr a i s the direc t su m o f simpl e ones . (39) Sho w tha t th e Li e algebr a g o f a compac t Li e grou p G i s alway s th e direc t sum o f it s cente r an d th e commutator : g = 3 0 [$,&]i an d th e commutato r i s semisimple. (40) Sho w tha t o n 5 2 , th e direc t su m o f th e tangen t bundl e wit h th e trivia l lin e bundle i s isomorphic t o the trivia l vecto r bundl e o f rank 3 (so in particular, th e 'cancellation law 5 doe s no t hol d fo r th e direc t su m o f vecto r bundles) .

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https://doi.org/10.1090/amsip/018/03

CHAPTER 2

Metric, Connection , an d Curvatur e

2.1. Metric , Connection , an d Curvatur e

Let M n b e a differentiabl e manifold , an d 7 r : E— * M a vector bundl e o f rank r over M. A metric o n £ i s a section g of S 2(E*) whic h is positive definite everywher e on M , i.e. , g(X,X) > 0 fo r an y 0 ^ X G E p a t an y p G M. I n othe r words , a metric g o n th e bundl e E assign s a n inne r produc t g p t o th e vecto r spac e E p fo r each p G M, suc h tha t g v depend s smoothl y o n p. B y th e loca l trivializatio n o f E and th e partitio n o f unity , i t i s eas y t o se e tha t ther e alway s exis t metric s o n an y given vecto r bundl e E. Next le t u s recal l th e definitio n o f a connection o n E. Writ e V = C°°(M) , an d denote b y A P(E) th e spac e o f all the section s o f the bundl e A^(TM* ) E. W e will call A P(E) th e spac e o f E-valued p- forms. DEFINITION 2.1 . A connection o n E i s a linea r ma p V : A°(E) - » A satisfying th e Leibniz ' rul e V(/£) = / V £ + d / ® £ , VteA°(E),

V/e

1

(E)

D

If X i s a vecto r field o n M , w e can evaluat e th e 1 -for m par t o f V £ o n X. Th e result (V£)(X ) i s a n elemen t o f A°(E). Fo r convenience , w e will writ e thi s sectio n as Vx£ , an d cal l i t th e covariant differentiation o f £ in th e directio n o f X. S o th e purpose o f having a connectio n o n E i s to enabl e u s t o differentiat e th e section s of E i n any give n tangen t direction . Not e that Vx £ i s P-linear i n the X position , bu t is onl y R-linea r i n th e £ positio n an d satisfie s th e Leibniz ' rul e whic h ca n no w b e written a s V x ( / £ ) = / V x £ + * ( / ) £ fo r an y / G £>, £ G 4 ° ( £) an d I G H . Again b y the partitio n o f unity, w e know that ther e alway s exist connection s o n E. Sinc e th e differenc e (V i — V2)x £ betwee n tw o connection s o f E i s D-linea r i n the £ position, i f we fix a connectio n V o o f E, the n th e spac e o f al l th e connection s on E i s simpl y V o -f A°(E* ® E TM*), a n infinit e dimensiona l affin e space . Given a connectio n V : A°(E) —> A 1 (E) o n E, w e ca n exten d i t t o a linea r map V p : A P(E)— > A v+l(E) fo r an y intege r p betwee n 1 and n — 1 by enforcin g the Leibniz ' rul e Vp(y>0 = d p f + ( - l ) V A V£, V

A 2(E) i s of particular importance . It i s called th e curvature tensor o f the connectio n V . Fo r an y sectio n £ of E and any functio n / o n M, we hav e

V2(/0 = Vi(/V £ + d/®0 - /V

2

(O + # A V£ + d ( d / ) 0 £ + ( - l )1# A V £ •

= /v 2 (0 So V 2 is a section o f the bundl e E* ® E A 2 (TM*), o r equivalently, a n End(E)valued 2-form . Tha t is , V2 G A2{End{E)). Evaluat e i t o n two vecto r field s X and y , w e get an endomorphism o f E, whic h wil l b e denoted b y \RXY* Clearly , RYX = —RXY- (W e will se e the reaso n o f this conspicuou s facto r ^ in a moment). Under a local fram e { e i , . . . , er} of E, writ e rr

Ve* = Y^ °ij ® eJ' V

2

(ei) = ^ 9 ^ 0 ej.

Then 0 = (%) , B = (&ij) ar e r xr matrices of local 1-forms or 2-forms, respectively . They ar e called the matrix of the connection or curvature of V under the local fram e e. B y th e definition , w e hav e 6 = old - 0 A 0.

We clai m tha t fo r any tw o vecto r fields X an d F o n M, i t hold s RXY =

V j V y - V y V x - V[x,y] -

In orde r t o verify that , le t e be a loca l fram e o f E, an d use th e same lette r e = *(ei,... ,e r ) t o denot e th e column vector . W e have V ^ e = 6(X)e a s a matri x equation. Unde r e , the endomorphis m RXY is represented b y the matri x 2©(X , F ). We hav e 2G(X, y) =

2(d0)(X , y) - 2(< 9 A 0)(X, Y)

= {x((9(y) ) - y(6i(x) ) - 6>([x , y])} - {0(x)0(y ) - e(Y)0(x)} = {x((?(y) ) +