On totally geodesic invariant submanifolds of an S-manifold

23 normal connection in S-manifolds whose invariant f-sectional L. M. FERNÁNDEZ i nvar i ant has defined [ 1 ]) , (Blair, for · an i nvar i ant submanifold of Blair, in an S-manifold of constant INTRODUCTIO N .- D. f-sectional curvature to be totally geodesic. codimension 2 obtained a condi tion of invariant submanifo lds of S-manifo ls. Specially , they have Tsuchiya , [4]) , have investigated some topics i n the geo metry [5] and Kon, [6]). Kobayashi and Tsuchiya, (Koba yashi and submanifolds of codimension greater that 2 and wi t h f lat The purpose of the present note is to study in var iant curvature is constant and to obtain a condition for them to be manifolds. On the other hand, many authors have studied invariant sub manifolds of Sasak ian manifolds, (see, e.g. , Kon , o. S-manifolds whi ch reduce, in a specia l case, to Sasakian I n [1] , S-m an i f o lds , w hic h r e du ce i n a s pecial case to Sasaki an m an i f ol ds , we r e de f i ne d . In t his not e, a co n ditio n for an i nvar ia nt s ubma nifol d of co di m~n s io n greater t ha n 2 i n a n S-mani- fol d to be t ota l ly geo desic i s o btai ne d . ON TOTALLY GEODESIC INVAR IANT SUB MA NIFOLDS OF ANSMANIFOLD Dpt o.: Alg ebra , Comp utació n, Ge omet rí a y To pología . Fa c ulta d de Mat emát i c as . Univ ers id ad de Sevilla . Apdo . de Cor r eos 1. 16 0. 410 80 SEVI LLA (Es paña) . Rev. Acad; Ci encias Zaragoza , 43 (1988) 24 The Gauss - Weingarten formulas are given by ~ is the Weingarten R(X,Y,U,V) = RO(X,y,U,V) - g([~,~]X,y), X,YET(M), U,VET(M).l, We denote by ~ the covariant differentiation in Nn and ~. (1. 2) given by endomorphism associated with V and it satisfies: where [~,~]X = ~X - ~X. g(~X,y) =g«(}'(X,Y) ,V). We denote by R, R and RO the curvature tensors associated with ~, 'V and ° respectively. If RO vanishes identically the normal connection ° is said to be flato The Ricci equation is ( 1.1) ~XV -~X +0XV, X,YeT(M), VET(M).l, where D is the connection in the normal bundle, a is the by 'V the covariant differentiation in Mm determined by the second fundamental form of Mm, l. PRELIMINARIES.- Let Nn be a Riemannian manifold of on induced metric. Let T(N) (resp. T(M» be the Lie algebra of vector fields on Nn (resp. on ~) and T (M).l the set of all vector fields normal to ~. dimension n and Mm an m-dimensional submanifold of N n. Let g be the metric tensor field on Nn as well as the induced metric section 2, definitions and sorne properties of S~manifolds. In surnmary of notations and formulas for submanifolds and, in totally geodesic. To this end, in section 1, we give a brief section 3we get the main resulto zero. are dual on g if a Riemannian metric such that, exists f-structure with F closed is called a g(X,Y) = g(fX ,fY) + ~(X,Y), X,YeT(N), fields A normal (2.2) F(X,Y) = g(X,fY), X,YeT(N). ~a(~~) = °a~i f~a =Di ~aof =Oi 2 f= -I +¿ ~a 0 ~a' a,~e{l, ... ,s}, a N2n+s is said to have an f-structure with complemented frames. is defined by K-structure and N2n+s is called a K-manifold. In such a 25 [f,f] + 2¿ ~ 0 d~ =O, a a a where [f, f] is the Nij enhuis torsion of f. Moreover, i t is Further, the f-structure is said to be normal if 1-forms, then manifold, the ~ are Killing vector fields, (Blair, [1]). . a Let ~ denote the distribution determined by _f 2 and ~ .the complementary distribution. ~ is determined by f2 + I and where ~(X,Y) =¿ ~D(X)~D(Y) . The fundamental 2-form F on N2n+s D satifying, (Yano, [7]): spanned by ~l' .. . '~s. If Xe~, then ~a(X) =O, for any a and if known that there (2.1) vector Finally, the submanifold MID is said to be totally geodesic in Nn if its second fundamental form is identically 2. S-MANIFOLDS. - Let N2n+s be a (2n+s) -dimensional manifold with an f-structure f of rank 2n. If there exist on N2n+s 26 invariant f-sectional curvature k, then its curvature tensor the N2n+s ., [ 1] ) . For the X, YeT(N) . an S-manifold on of g -fX, XeT(N), ae{l, ... ,s}, = L[g(fX,fY)~a + ~a(Y)f2xJ, a . connection ~ - 2F(X,Y)F(Z,W)}, X,Y,Z,WeT(N). R(X,y,Z,W) =¿ {g(fX,fW)~a(Y)~~(Z) - a,~ - g(fX,fZ)~a(Y)~~(W) + g(fy,fZ)~a(X)~~(W) - - g(fy,fW)~a(X)~~(Z)} + + «1/4) (k+3s) {g(X,W)g(fY,fZ) - g(X,Z)g(fY,fW) + + g(fY,fW)~(X,Z) - g(fY,fZ)~(X,W)} + + (1/4) (k-s) {F(X,W)F(Y,Z) - F(X,Z)F(Y,W) - In the case s= 1, an S-manifold is a Sasakian manifold. (2.5) with certain conditions is an S-manifold. In this way, a space . of a principal toroidal bundle over a Kaehler manifold (Blair, [2]), (Blair, Ludden and Yano, [3]). Thus, the bundle For s~2, examples of S-manifolds are gi ven in (Blair, [1] ) , has the form, (Kobayashi and Tsuchiya, [4]) an orthonormal pair spanning the section. The sectional Aplane section rr is called an invariant f-section if it 2n+s is determined by a vector Xe~(p), peN , such that {X,fX} is curvature K(X, fX), denoted by H(X), is called an invariant f-sectional curvature. If N2n+s is an S-manifold of constant following were also proved: Riemannian manifolds have been studied in (Blair, A K-structure such that F= d~a' a= 1, ... ,s, is called an S-structure and N2n+ s is called an S-manifold. These (2.3) (2.4) XeM, then fX =O. 27 an invariant submanifold of an S-manifold is such that an fX =TX + NX, (2.6) (2.7) and, so, m= 2p+s. For later use, we prove the following Lemma 2.1.- Let M 2p+s be an invariant submanifold of Proof: By using the Weingarten formula (1.1), (2.4) and the fact that M m+s is an invariant submanifold, it is easy to show that AfVX = f~X. Now, if YeT(M), we have g(AfVX,y) = g(X,AfVY) = g(X,f~Y) = -g(~fX,y) l l fVeT(M) , for any VeT(M) . Moreover, it is an S-manifold too . 2n+s l S-man~fold N. Then, for any XeT(M), VeT(M) , ae{l, ... ,s}: it defines an f-structure in the tangent bundle. On the other hand, if one of the ~a is normal to Mm, then T=O, because g(X,fY) = F(X,Y) = d~a(X,y) =O, X,YeT(M), The submanifold ~ is said to be invariant if all of ~ a (a 1, ... ,s) are always tangent to ~ and N is Lderrt Lca Lt y zero, i.e., fXeT(M), for any XeT(M). It is easy to show that tangent bundle. It is easy to show that if T does not vanish, normal component of fX. Then, T is an endomorphism of the tangent bundle and N is a normal-bundle valued 1-form on the where TX is the tangential component of fX and NX is the introduced as a canonical example of an S-manifold playing the role of complex projective space in Kaehler geometry and the odd-dimensional sphere in Sasakian geometry. Now, let ~ be an m-dimensional submanifold immersed in an S-manifold N2n+ s. For any XeT(M), we write generalization of the Hopf fibration rr':S2n+1~~~n is 28 Ricci equation (1.2) and Lemma 2.1, we have and (2.7) holds. an Let M 2p+s be such that the normal (s-k)g(fX,fY) = 4g(~X,~y). Now, we prove Theorem 3.2. - Let M2p+s be an invariant submanifold of an . 2n+s 2p+s S-man~fold N (k). If the codimension of M is greater than 2, then the normal con~ection of M2p+s is flat if and only if k=s and M 2p+s is totally geodesic. Proof: From the Ricci equation (1.2) and (2.5), i t is clear that if M 2p+s is totally geodesic, then its normal connection is flato Now, we suppose that M 2p+s is not totally geodesic. We can choose a local field of orthonormal frames for vector fields in M 2p+s in the form R(X,fy,V,fV) = 2g(~X,~y), for any vector field X,YeT(M) and any unit vector field V~T(M)i. Now, from (2.5) we obtain Then, we get the following proposition 3.1.- Let M2p+s be an invariant submanifold of an S-manifold N2n+ s(k) with flat normal connection. Then, k~s and the equality holds if and only if M2p+s is totally geodesic. (3.1) invariant submanifold of N2n+ s(k) connection of M 2p+s is flat, i. e., RD=O. Then, by using the f-sectional curvature is a constant k. 3 . INVARIANT SUBMANIFOLDS WITH FLAT NORMAL CONNECTION. - In this section, let N2n+s (k) be an . S-manifold whose invariant 29 is of constant invariant f -sectional curvature. again, that ~ + ~ =O, and fV. Regarding to (3.3), it follows that g(~X,~y) = O, for any X,YeT(M). Consequently, the vector fields ~(E1)'" .,~(E2p) ,~(E1)'·· · ,~(E2p) are linearly independent, which "is a contradiction. Therefore, M 2p+s is totally geodesic and k= s. invariant submanifold of codimension 2 in an S-manifold N2n+ s(k). Then, 1f"l+s is totally geodesic if and only if 1f"l+s Finally, for codiroension 2, we have the following Theorero 3.3. - (Kobayashi and Tsuchiya, [4]) Let 1f"l+s be an " ~ for any orthonorroal vector fields V,WeT(M) . Thus, froro (1.2) and (3.2), we get R(X,y,V,W) = 2g(~X,~y), X,YeT(M). Using (2.5), we obtain {E 1,···,E p,E p+1= fE 1,···,E 2p =fE p'€l'···'€S }· If ~(Ei) =O, for soroe unit vector field VeT(M)~, the ~, froro (3.1), we get that M 2p+s is totally geodesic, by virtue of Proposition 3.1. Thus, ~(Ei) ~ O, for any Ei and V, and so, ~(E1), ... ,~(E2p) are linearly independent. on the other hand, it is easy to show, by using (3.1) (3 .3) (s-k)g(X,fY)g(V,fW) = 4g(~X,~y). If the codiroension of M 2p+s is greater than 2, we can take a unit vector field W in T(M)~ which is , orthogonal to V (3.2) [2] REFERENCES [1] Blair, O.E.: 1970, J. oiff. Geom. ~, (155-167). : 1971, An. ti. Univ. "Al. l. Cuza" la i, (Serie nova) XVII, Fase. ~, (171-177). [3] Blair, O.E., Ludden, G.O. and Yano, K.: 1973, Trans. ' Am . Math. Soco 181, (175-184). [4] Kobayashi, M. and Tsuchiya, S.: 1972, Kodai Math. Sem. Rep. 24, ( 430 ..: 45 ~) . [5] Kon, M.: 1973, Kodai Math. Sem. Rep. 25, (330-336). [6] 1974, Tensor N.S. 28, (133-138). [7] Yano, K. : 1963, Tensor 14, (99-109). 30