A Rigidity Theorem forManifold with a N ice Subman

应

用

数

学

M A TH EM A T ICA A PPL ICA TA 1999, 12 ( 1) : 72～ 75

A R ig id ity Theorem for M an ifold 3 w ith a N ice Subman ifold Ξ

Xu Sen lin ( 徐森林) 　H uang Zheng ( 黄正) 　Q i F eng ( 祁锋)

(D ep t. of M a th. U n iv ersity of S cience and T echnology of C h ina , H ef ei 230026) Abstract　 In th is p ap er, w e p rove a rigidity theo rem fo r a com p lete R iem ann ian m an ifo ld by the ex istence of a n ice to ta lly geodesic subm an ifo ld. T hen w e sta te a co ro lla ry w h ich ha s the fo rm of the w ell2know n Bonnet 2M yers theo rem. W e a lso po in t ou t tha t in ou r m a in theo rem , " to ta lly geodesic" can no t be taken p lace by "m in im a l" in ou r w ay.

Keywords: K 2th R icci cu rva tu re; To ta lly geodesic subm an ifo ld; R ig id ity theo rem AM S ( 1991) Subject Cla ss if ica tion: 53C21 L et M be an m 2d im en siona l com p lete R iem ann ian m an ifo ld, the Bonnet 2M yers theo rem [ 1, 4 ] sta tes tha t if K M ≥c o r fo r a ll un it V ecto rs X , R ic (X , X ) ≥ ( n - 1 ) c, w here c > 0 then every geodesic of leng th Πg c ha s con juga te po in t s. H ence M is com p act and d (M ) , the d i2 am eter of M , sa t isfies d (M ) ≤Πg c . Since to ta lly geodesic subm an ifo ld s a re the h igher d i2 m en siona l genera liza t ion of geodesics, it is rea sonab le fo r u s to exp ect tha t the ex istence of a certa in k ind of n ice to ta lly geodesic subm an ifo ld s in such a m an ifo ld M m ay a lso g ives som e rig id ity p henom enon. A n a rgu rm en t of the second va ria t ion of a rc2leng th g ives u s m a in theo 2 rem of th is p ap er a s fo llow s M a in Theorem 　 let M be an m 2d im en siona l com p lete connected R iem ann ian m an ifo ld n ’ and P is an a rb ita ry po in t of M . If w e have K (H t , Χ ( t) ) ≥ nc > 0 a long each m in im izing n ’ ⊥ ’ ⊥ geodesic Χ: [ 0, 1 ] →M sta rt ing from p fo r any n 2p lane H 0 < Χ ( 0) , w here Χ ( 0) deno tes the n ’ o rthogona l com p lem en t of Χ ( 0) in T pM and H t deno tes the p a ra llel t ran sla te of the p lane H

n

0

< T pM th rough P to Χ( t) a long Χ. If M ha s an im m ersed com p act to ta lly geodesic subm an i2 fo ld N w ho se d im en sion r ≥n , then the m an ifo ld M is com p act. T he fo llow ing theo rem is an im m ed ia te con sequence of ou r m a in theo rem , w e w rite it dow n in the fo rm of the Bonnet 2M yers theo rem Theorem 　L et M be an m 2d im en siona l com p lete connected R iem ann ian m an ifo ld w ith the k 2th R icci cu rva tu re R ickM ≥ kc > 0. If M ha s a com p act im m ersed to ta lly subm an ifo ld w ho se d im en sion ≥ k , then M is com p act. Ξ R eceived data: 1998204223 P ro ject Suppo rted by NN SFC and N ECYSFC

N o. 1

A R igidity T heo rem fo r M an ifo ld w ith a N ice Subm an ifo ld

73

L et e, e1 , …, ek ∈ T PM be any ( k + 1) 2m u tua lly o rthogona l un it tangen t vecto rs, w here p is any po in t of M . W e w rite K ( e ∧ e i ) fo r the sect iona l cu rva tu re of the p lane sp anned by e and e i ( 1≤ i≤k ). W e define R ickM > sup e i, e K (e i ∧e) a s the k 2th R icci cu rva tu re of M . L et p ∈ M , H < T pM be an r2p lane sp anned by r o rthono rm a l tangen t vecto rs e1 , e2 …, e r ∈ T pM and X r

∈ T pM be a tangen t vecto r o rthogona l to H r. W e deno te K (H r , X ) = 6 i= 1 K ( e i ∧ X ). Cer2 r ta in ly, it is ea sy to verify tha t K (H , X ) is indep erden t of cho ice of the vecto rs e1 , …, e r. In r fact let c1 , c2 , …, c r be o rthono rm a l tangen t vecto rs w h ich sp an H , then there ex ist Κij ∈R such r

tha t c i = r

6

i= 1

6

r j= 1

r

Κij e j , w here ( Κij )

r

K

6

is an o rthogona l m a t rix, thu s,

Κij e j ∧ X

K (ci ∧ X ) =

i= 1

r

j= 1

6

r

6

6

1 1 = Κij ΚikR ( e j , X , ek , X ) = ∆jkR ( e j , X , ek , X ) = 〈X , X 〉i, j , k = 1 〈X , X 〉j , k = 1

r

6

K ( e i ∧ X ). w here ∆jk is K ronecker funct ion .

i= 1

Proof of M a in Them orem 　 F irst ly w e fix p ∈M , since the subm an ifo ld N is com p act, there ex ist s a po in t p 1 of N such tha t d > d ( p , p 1 ) = d ( p , N ). L et Χ( t) , t ∈ [ 0, d ] be a m in i2 m izing geodesic in M p a ram et rized by a rc2leng th from p to p 1 such tha t Χ rea lizes the d istance from p to N . Cla im 1　L et N be a subm an ifo ld of M , w ithou t bounda ry, Χ: [ 0, l ] → M be a geodesic ’ such tha t Χ( 0) | N , Χ( l ) = p 1 ∈ N . If Χ is the sho rtest cu rve from Χ( 0) to N , then Χ ( l ) is p erp end icu la r to T p 1N . ’ Proof of Cla im 1　 If Χ ( l ) is no t p erp end icu la r to T p 1N , w e choo se Y ∈T P 1N such tha t〈 ’ Χ ( l ) , Y 〉> 0. L et c be a cu rve in N sta rt ing from p 1 w ith in it ia l tangen t vecto r Y. T hu s w e ζ can con st ruct a va ria t ion Α: [ 0, l ] × [ - Ε, Ε] →M such tha t Αg [ 0, l ] ×{0} = Χ, Α( 0, s ) = c ( s) , Α( l,

s) = Χ( 0) . W e deno te Χs = Αg [ 0, l ] ×s , V ( t, s ) =

5Α( ) 5Α( ) t, s , T ( t, s ) = t, s . F rom the first va ria 2 5s 5t

t ion fo rm u la w e have l

〈g V , T 〉 dt ∫ = gΧg ∫(T〈V , T 〉- 〈V , g T 〉) d t = g Χ g 〈V , T 〉 g - 〈V , g T 〉 dt , ∫

d ’ L [ Χs ]s= 0 = g Χ g ds

1

’ - 1

T

0 l

T

0

l

’ - 1

l

0

T

0

ζ ζ w here Χ is Χ it self, bu t the d irect ion is from Χ( l ) to Χ( 0 ). N o te tha t Χ = Χ0 is a geodesic and Α( l , s) ≡ Χ( 0) , thu s the term s g T T g s= 0 and〈V , T 〉 g s= l van ish. So , d ’ - 1 1 λ’ ( 0) , Y 〉 L [ Χs ] s= 0 = - g Χ g 〈V , T 〉 g s= 0 = - g Χ’g - 〈Χ ds 1 λ’ = - g Χ’g - 〈Χ 〉( l ) , Y > < 0.

T herefo rre, fo r sm a ll s, L [ Χs ] < L [ Χ], w h ich con t rad ict s the a ssum p t ion tha t Χ is the sho rtest cu rve from Χ( 0) to N . T h is com p letes ou r p roof of cla im 1. N ow w e take an o rthono rm a l ba sis e1 , …, e r of T P 1N and let{E i ( t) } be the p a ra llel t ran s2 la te of {e i } to Χ( t) a long Χ. Set W i ( t) =

sin

Πt 2d

E i ( t) , 1 ≤ i ≤ r be vecto r field on Χ. Each W

i

74

1999

M A TH EM A T ICA A PPL ICA TA

( t) g ives rise to a geodesic va ria t ion of the va ria t iona l cu rves of the geodesic Χ( t ) , keep ing one endpo in t p fixed and o ther endpo in t s on shbm an ifo ld N , since w e a ssum e a t first tha t N is to ta lly geodesic. L et L i be the a rc2leng th of the va ria t iona l cu rve w h ich w e induced by W t ( t). ’ ″ T hu s L i ( 0) = 0, w e can com p u te L i ( 0) by u sing the second va ria t ion fo rm u la of a rc2leng th.

L et g and R a s the L evi- C ivita connect ion and the cu rva tu re ten so r of M , resp ect ively. Suppo se g is the R iem ann ian connect ion of N . Ea sy of see tha t the second fundam en ta l fo rm of N is zero , hence the tangen t vecto r field a long the geodesics in N a re ju st a long the sam e ″ geodesics in M . B efo re w e com p u te L i , w e need to do som e p rep a ra t ion. F irst ly, w e say tha t 〈W i , Χ’〉( t) = 0 . T h is fact can be ea sily seen since w e know tha t W i ( 0) = 0 and〈W i , Χ’〉( d ) = 0. Second ly, w e say tha t〈Χ’( t) , g W W g d0 = 0. In fact, i〉 i

( d ) - < Χ’( t) , g W W ( 0) 〈Χ’( t) , g W W g d0 = 〈Χ’( t) , g W W i〉 i > i〉 i i i ( d ) = 〈Χ’( t) , g W W (d ) = 0 = 〈Χ’( t) , g W W i〉 i〉 i i here, w e no te tha t W i ( 0) = 0. T hu s, w e have, d

∫(g g

″ ’ L i ( 0) =〈Χ ( t) , g W W g d0 + i〉 i

0

W i Χ’

g 2)

’2 ) dt - 〈W i , Χ’〉 - 〈R ( Χ’, W i )W i , Χ’〉 d

∫(g g = ∫(g g =

0

W i Χ’

’2 ’ ’ ) dt g 2 - 〈W i , Χ’〉 - 〉 R ( Χ , W i )W i , Χ 〉

W i Χ’

g 2 - < R ( Χ’, W i )W i , Χ’> ) d t

d

0

″ B y add ing a ll L i ( 0) from i = 1 to i = r, w e have r

6

d

r

0

i= 1

∫6

″ L i ( 0) =

i= 1

( g g Χ’W i g 2 - 〈R ( Χ’, W i )W i , Χ’〉 d t

Π Πt co s 2d 2d

r

∫6 d

=

0

i= 1

r

2

6

-

sin

i= 1

Πt 2 ( ’ ) 〈R Χ , E i E i , Χ’〉 d t 2d

Πt Πt Π co s2 - sin 2 〈R ( Χ’, E i ) E i , Χ’〉 d t. 0 2d 2d 6 4d 2 i= 1 n ’ R eca ll tha t K (H t , Χ ( t) ) nc, fo r i1 , i2 , …, in ∈ {1, …, r }, i i ≠ ik ( j ≠ k ) , r ≥ n , w e d

∫

=

r

2

r

n

have

6 〈R ( Χ , E ’

ij

( t) ) E ij ( t) , Χ’〉≥ nc. T herefo re, w e have

j= 1

r

r

1 6 〈R ( Χ’, E i ) E i , Χ’〉= n 6 〈R ( Χ’, E i ) E i , Χ’〉 n

j= 1

=

≥ 　　Cla im 2　 d ( p , N ) ≤ Πg 2

6

i= 1

″ L i ( 0) =

d

n C nr

n

6

6 〈R ( Χ , E ’

1≤ i i < …< in ≤ r j = 1

6

r n nC r 1≤ i1 < …
Πg r

i= 1

r

N o. 1

A R igidity T heo rem fo r M an ifo ld w ith a N ice Subm an ifo ld

75

d Πt Πt Πt - sin 2 d t = rc co s d t = 0. 0 0 2d 2d d ″ T herefo re, there ex ist s som e i, 1 ≤ i ≤ r , such tha t L i ( 0) < 0, w h ich con t rad ict s the fact tha t Χ rea lizes the sho rtest d istance from p to N . T h is com p letes the p roof of cla im 2. 2 c , becau se the po in t N ow , fo r any po in t q o ther than p in M , w e have d ( q , N ) ≤ Πg p above is a ssum ed to be an a rb ita ry po in t in M . J u st like befo re, w e choo se q 1 ∈ N such tha t d ( q , q 1 ) = d ( q , N ) , So d ( p , q ) ≤d ( p , p 1 ) + d ( p 1 , q 1 ) + d ( q 1 , q ) d

∫

< rc

∫

co s2

≤Πg( 2

(2 c ) + d 0 + Πg

c ) = Πg

c + d 0,

w here d 0 deno tes the d iam eter of the com p act m an ifo ld N . N ow d (M ) ≤ d 0 + Πg c . H ence M is com p act, w h ich com p letes the p roof of ou r m a in theo rem.

In the ca se tha t T heo rem 2 sta tes, w e can p rove tha t d ( p , N ) is bounded in the sam e w ay a s the p roof of m a in theo rem. Sim ilia rly, w e can ob ta in tha t M is com p act. Remark 　 If N is ju st a m in im a l subm an ifo ld bu t no t a to ta lly geodesic subm an ifo ld, m a in theo rem in th is p ap er m ay be w rong. R eca ll tha t each W i g ives rise to a va ria t ion of the va ria t iona l cu rves of Χ keep ing p fixed and o ther endpo in t s on N . If N is ju st m in im a l, geodesic in N m ay no t begeodesic in M , and then o ther endpo in t s m ay no t no N , th is m akes ″ ou r com p u t ing L i ( 0) inva lid. References 1 　Cheeger J and Eb in D. Com p a rison T heo rem s In R iem ann ian Geom etry, N o rth 2Ho lland, Am sterdam , 1975. 2　K lingenberg W. R iem ann ian Geom etry, B erlin, N ew Yo rk, de G ruyter, 1982. 3 　Kobaya sh i S and N om izu K. Founda tion s O f D ifferen tia l Geom etry, V o l. 1, In terscience, N ew Yo rk, 1963: V o l. 2, In terscience, N ew Yo rk, 1969. 4 　M ayers S. Cu rva tu re O f C lo sed H yp ersu rfaces A nd N on 2ex istence O f C lo sed M in im a l H yp ersu rfaces, T ran s. Am er. M a th. Soc. 1951, 71: 211～ 217 5　W u H. M an ifo lds O f Pa rtia lly Po sitive Cu rva tu re, Indiana U. M a th. Jou rna l, 1987, 36 ( 3) :

一类流形的刚性定理 徐森林　黄　正　祁　锋 ( 中国科技大学数学系　安徽合肥 230026)

摘　要 本文通过具有良好性质的子流形的存在性, 证明了一类流形的一个刚性定理, 并得到形如 Bonnet 2M yers 定理的推论. 我们还指出, 在主要定理中全测地子流形的条件一般不能减弱为 极小子流形. 关键词: 第 k 个 R icci 曲率; 全测地子流形; 刚性定理