On totally geodesic invariant submanifolds of an S-manifold

23 no mal connec ion in S-mani olds whose in a ian -sec ional L. M. FERNÁNDEZ i n a i an has de ined [ 1 ]) , (Blai , o · an i n a i an submani old o Blai , in an S-mani old o cons an INTRODUCTIO N .- D. -sec ional cu a u e o be o ally geodesic. codimension 2 ob ained a condi ion o in a ian submani o lds o S-mani o ls. Specially , hey ha e Tsuchiya , [4]) , ha e in es iga ed some opics i n he geo me y [5] and Kon, [6]). Kobayashi and Tsuchiya, (Koba yashi and submani olds o codimension g ea e ha 2 and wi h la The pu pose o he p esen no e is o s udy in a ian cu a u e is cons an and o ob ain a condi ion o hem o be mani olds. On he o he hand, many au ho s ha e s udied in a ian sub mani olds o Sasak ian mani olds, (see, e.g. , Kon , o. S-mani olds whi ch educe, in a specia l case, o Sasakian I n [1] , S-m an i o lds , w hic h e du ce i n a s pecial case o Sasa- ki an m an i ol ds , we e de i ne d . In his no e, a co n di io n o an i n a ia n s ubma ni ol d o co di m~n s io n g ea e ha n 2 i n a n S-mani- ol d o be o a l ly geo desic i s o b ai ne d . ON TOTALLY GEODESIC INVAR IANT SUB MA NIFOLDS OF ANS- MANIFOLD Dp o.: Alg eb a , Comp u ació n, Ge ome í a y To pología . Fa c ul a d de Ma emá i c as . Uni e s id ad de Se illa . Apdo . de Co eos 1. 16 0. 410 80 SEVI LLA (Es paña) . Re . Acad; Ci encias Za agoza , 43 (1988) 24 The Gauss - Weinga en o mulas a e gi en by ~ is he Weinga en R(X,Y,U,V) = RO(X,y,U,V) - g([~,~]X,y), X,YET(M), U,VET(M).l, We deno e by ~ he co a ian di e en ia ion in Nn and ~. (1. 2) gi en by endomo phism associa ed wi h V and i sa is ies: whe e [~,~]X = ~X - ~X. g(~X,y) =g«(}'(X,Y) ,V). We deno e by R, R and RO he cu a u e enso s associa ed wi h ~, 'V and ° espec i ely. I RO anishes iden ically he no mal connec ion ° is said o be la o The Ricci equa ion is ( 1.1) ~XV -~X +0XV, X,YeT(M), VET(M).l, whe e D is he connec ion in he no mal bundle, a is he by 'V he co a ian di e en ia ion in Mm de e mined by he second undamen al o m o Mm, l. PRELIMINARIES.- Le Nn be a Riemannian mani old o on induced me ic. Le T(N) ( esp. T(M» be he Lie algeb a o ec o ields on Nn ( esp. on ~) and T (M).l he se o all ec o ields no mal o ~. dimension n and Mm an m-dimensional submani old o N n. Le g be he me ic enso ield on Nn as well as he induced me ic sec ion 2, de ini ions and so ne p ope ies o S~mani olds. In su nma y o no a ions and o mulas o submani olds and, in o ally geodesic. To his end, in sec ion 1, we gi e a b ie sec ion 3we ge he main esul o ze o. a e dual on g i a Riemannian me ic such ha , exis s -s uc u e wi h F closed is called a g(X,Y) = g( X , Y) + ~(X,Y), X,YeT(N), ields A no mal (2.2) F(X,Y) = g(X, Y), X,YeT(N). ~a(~~) = °a~i ~a =Di ~ao =Oi 2 = -I +¿ ~a 0 ~a' a,~e{l, ... ,s}, a N2n+s is said o ha e an -s uc u e wi h complemen ed ames. is de ined by K-s uc u e and N2n+s is called a K-mani old. In such a 25 [ , ] + 2¿ ~ 0 d~ =O, a a a whe e [ , ] is he Nij enhuis o sion o . Mo eo e , i is Fu he , he -s uc u e is said o be no mal i 1- o ms, hen mani old, he ~ a e Killing ec o ields, (Blai , [1]). . a Le ~ deno e he dis ibu ion de e mined by _ 2 and ~ . he complemen a y dis ibu ion. ~ is de e mined by 2 + I and whe e ~(X,Y) =¿ ~D(X)~D(Y) . The undamen al 2- o m F on N2n+s D sa i ying, (Yano, [7]): spanned by ~l' .. . '~s. I Xe~, hen ~a(X) =O, o any a and i known ha he e (2.1) ec o Finally, he submani old MID is said o be o ally geodesic in Nn i i s second undamen al o m is iden ically 2. S-MANIFOLDS. - Le N2n+s be a (2n+s) -dimensional mani old wi h an -s uc u e o ank 2n. I he e exis on N2n+s 26 in a ian -sec ional cu a u e k, hen i s cu a u e enso he N2n+s ., [ 1] ) . Fo he X, YeT(N) . an S-mani old on o g - X, XeT(N), ae{l, ... ,s}, = L[g( X, Y)~a + ~a(Y) 2xJ, a . connec ion ~ - 2F(X,Y)F(Z,W)}, X,Y,Z,WeT(N). R(X,y,Z,W) =¿ {g( X, W)~a(Y)~~(Z) - a,~ - g( X, Z)~a(Y)~~(W) + g( y, Z)~a(X)~~(W) - - g( y, W)~a(X)~~(Z)} + + «1/4) (k+3s) {g(X,W)g( Y, Z) - g(X,Z)g( Y, W) + + g( Y, W)~(X,Z) - g( Y, Z)~(X,W)} + + (1/4) (k-s) {F(X,W)F(Y,Z) - F(X,Z)F(Y,W) - In he case s= 1, an S-mani old is a Sasakian mani old. (2.5) wi h ce ain condi ions is an S-mani old. In his way, a space . o a p incipal o oidal bundle o e a Kaehle mani old (Blai , [2]), (Blai , Ludden and Yano, [3]). Thus, he bundle Fo s~2, examples o S-mani olds a e gi en in (Blai , [1] ) , has he o m, (Kobayashi and Tsuchiya, [4]) an o hono mal pai spanning he sec ion. The sec ional Aplane sec ion is called an in a ian -sec ion i i 2n+s is de e mined by a ec o Xe~(p), peN , such ha {X, X} is cu a u e K(X, X), deno ed by H(X), is called an in a ian -sec ional cu a u e. I N2n+s is an S-mani old o cons an ollowing we e also p o ed: Riemannian mani olds ha e been s udied in (Blai , A K-s uc u e such ha F= d~a' a= 1, ... ,s, is called an S-s uc u e and N2n+ s is called an S-mani old. These (2.3) (2.4) XeM, hen X =O. 27 an in a ian submani old o an S-mani old is such ha an X =TX + NX, (2.6) (2.7) and, so, m= 2p+s. Fo la e use, we p o e he ollowing Lemma 2.1.- Le M 2p+s be an in a ian submani old o P oo : By using he Weinga en o mula (1.1), (2.4) and he ac ha M m+s is an in a ian submani old, i is easy o show ha A VX = ~X. Now, i YeT(M), we ha e g(A VX,y) = g(X,A VY) = g(X, ~Y) = -g(~ X,y) l l VeT(M) , o any VeT(M) . Mo eo e , i is an S-mani old oo . 2n+s l S-man~ old N. Then, o any XeT(M), VeT(M) , ae{l, ... ,s}: i de ines an -s uc u e in he angen bundle. On he o he hand, i one o he ~a is no mal o Mm, hen T=O, because g(X, Y) = F(X,Y) = d~a(X,y) =O, X,YeT(M), The submani old ~ is said o be in a ian i all o ~ a (a 1, ... ,s) a e always angen o ~ and N is Lde Lca L y ze o, i.e., XeT(M), o any XeT(M). I is easy o show ha angen bundle. I is easy o show ha i T does no anish, no mal componen o X. Then, T is an endomo phism o he angen bundle and N is a no mal-bundle alued 1- o m on he whe e TX is he angen ial componen o X and NX is he in oduced as a canonical example o an S-mani old playing he ole o complex p ojec i e space in Kaehle geome y and he odd-dimensional sphe e in Sasakian geome y. Now, le ~ be an m-dimensional submani old imme sed in an S-mani old N2n+ s. Fo any XeT(M), we w i e gene aliza ion o he Hop ib a ion ':S2n+1~~~n is 28 Ricci equa ion (1.2) and Lemma 2.1, we ha e and (2.7) holds. an Le M 2p+s be such ha he no mal (s-k)g( X, Y) = 4g(~X,~y). Now, we p o e Theo em 3.2. - Le M2p+s be an in a ian submani old o an . 2n+s 2p+s S-man~ old N (k). I he codimension o M is g ea e han 2, hen he no mal con~ec ion o M2p+s is la i and only i k=s and M 2p+s is o ally geodesic. P oo : F om he Ricci equa ion (1.2) and (2.5), i is clea ha i M 2p+s is o ally geodesic, hen i s no mal connec ion is la o Now, we suppose ha M 2p+s is no o ally geodesic. We can choose a local ield o o hono mal ames o ec o ields in M 2p+s in he o m R(X, y,V, V) = 2g(~X,~y), o any ec o ield X,YeT(M) and any uni ec o ield V~T(M)i. Now, om (2.5) we ob ain Then, we ge he ollowing p oposi ion 3.1.- Le M2p+s be an in a ian submani old o an S-mani old N2n+ s(k) wi h la no mal connec ion. Then, k~s and he equali y holds i and only i M2p+s is o ally geodesic. (3.1) in a ian submani old o N2n+ s(k) connec ion o M 2p+s is la , i. e., RD=O. Then, by using he -sec ional cu a u e is a cons an k. 3 . INVARIANT SUBMANIFOLDS WITH FLAT NORMAL CONNECTION. - In his sec ion, le N2n+s (k) be an . S-mani old whose in a ian 29 is o cons an in a ian -sec ional cu a u e. again, ha ~ + ~ =O, and V. Rega ding o (3.3), i ollows ha g(~X,~y) = O, o any X,YeT(M). Consequen ly, he ec o ields ~(E1)'" .,~(E2p) ,~(E1)'·· · ,~(E2p) a e linea ly independen , which "is a con adic ion. The e o e, M 2p+s is o ally geodesic and k= s. in a ian submani old o codimension 2 in an S-mani old N2n+ s(k). Then, 1 "l+s is o ally geodesic i and only i 1 "l+s Finally, o codi oension 2, we ha e he ollowing Theo e o 3.3. - (Kobayashi and Tsuchiya, [4]) Le 1 "l+s be an " ~ o any o hono oal ec o ields V,WeT(M) . Thus, o o (1.2) and (3.2), we ge R(X,y,V,W) = 2g(~X,~y), X,YeT(M). Using (2.5), we ob ain {E 1,···,E p,E p+1= E 1,···,E 2p = E p'€l'···'€S }· I ~(Ei) =O, o so oe uni ec o ield VeT(M)~, he ~, o o (3.1), we ge ha M 2p+s is o ally geodesic, by i ue o P oposi ion 3.1. Thus, ~(Ei) ~ O, o any Ei and V, and so, ~(E1), ... ,~(E2p) a e linea ly independen . on he o he hand, i is easy o show, by using (3.1) (3 .3) (s-k)g(X, Y)g(V, W) = 4g(~X,~y). I he codi oension o M 2p+s is g ea e han 2, we can ake a uni ec o ield W in T(M)~ which is , o hogonal o V (3.2) [2] REFERENCES [1] Blai , O.E.: 1970, J. oi . Geom. ~, (155-167). : 1971, An. i. Uni . "Al. l. Cuza" la i, (Se ie no a) XVII, Fase. ~, (171-177). [3] Blai , O.E., Ludden, G.O. and Yano, K.: 1973, T ans. ' Am . Ma h. Soco 181, (175-184). [4] Kobayashi, M. and Tsuchiya, S.: 1972, Kodai Ma h. Sem. Rep. 24, ( 430 ..: 45 ~) . [5] Kon, M.: 1973, Kodai Ma h. Sem. Rep. 25, (330-336). [6] 1974, Tenso N.S. 28, (133-138). [7] Yano, K. : 1963, Tenso 14, (99-109). 30