RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS

Transcription

1 J. Austral. Math. Soc. 72 (2002), RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS ION MIHAI (Received 5 June 2000; revised 19 February 2001) Communicated by K. Wysocki Abstract Recently, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Afterwards, we dealt with similar problems for submanifolds in complex space forms. In the present paper, we obtain sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. Also, estimates of the scalar curvature and the k-ricci curvature respectively, in terms of the squared mean curvature, are proved Mathematics subject classification: primary 53C0, 53B25, 53C25. Keywords and phrases: Ricci curvature, mean curvature, Sasakian space form, C-totally real submanifold, contact CR-submanifold.. 1. Preliminaries A./-dimensional Riemannian manifold. M ; g/ is said to be a Sasakian manifold if it admits an endomorphism of its tangent bundle T M, a vector field ¾ and a 1-form, satisfying: 2 = Id + ¾;.¾/ = 1; ¾ = 0; = 0; g.x;y / = g.x; Y /.X/.Y /;.X/ = g.x;¾/;. X /Y = g.x; Y /¾ +.Y /X; X ¾ = X; for any vector fields X; Y on T M,where denotes the Riemannian connection with respect to g. c 2002 Australian Mathematical Society /2000 $A2:00 + 0:00 27

2 28 Ion Mihai [2] A plane section ³ in T p M is called a -section if it is spanned by X and X,where X is a unit tangent vector orthogonal to ¾. The sectional curvature of a -section is called a -sectional curvature. A Sasakian manifold with constant -sectional curvature c is said to be a Sasakian space form and is denoted by M.c/. The curvature tensor of M.c/ of a Sasakian space form M.c/ isgivenby[1] (1.1) R.X; Y /Z = c + 3 c 1 {g.y; Z/X g.x; Z/Y }+ {.X/.Z/Y.Y /.Z/X + g.x; Z/.Y /¾ g.y; Z/.X/¾ + g.y; Z/X g.x; Z/Y 2g.X; Y /Z}; for any tangent vector fields X; Y; Z on M.c/. As examples of Sasakian space forms we mention R and S, with standard Sasakian structures (see [1]). Let M be an n-dimensional submanifold of a Sasakian space form M.c/ of constant -sectional curvature c. We denote by K.³/ the sectional curvature of M associated with a plane section ³ T p M, p M, and the Riemannian connection of M, respectively. Also, let h be the second fundamental form and R the Riemann curvature tensor of M. Then the equation of Gauss is given by (1.2) R.X; Y; Z; W / = R.X; Y; Z; W / + g.h.x; W /; h.y; Z// g.h.x; Z/; h.y; W //; for any vectors X, Y, Z, W tangent to M. Let p M and {e 1 ;:::;e n } an orthonormal basis of the tangent space T p M.We denote by H the mean curvature vector, that is (1.3) We also set H.p/ = 1 n h.e i ; e i /: (1.) and (1.5) h r ij = g.h.e i ; e j /; e r / h 2 = g.h.e i ; e j /; h.e i ; e j //: i; j=1 For any tangent vector field X to M, we put X = PX+ FX,wherePX and FX are the tangential and normal components of X, respectively. We write (1.6) P 2 = g 2.Pe i ; e j /: i; j=1

3 [3] Ricci curvature of submanifolds 29 Suppose L is a k-plane section of T p M and X a unit vector in L. We choose an orthonormal basis {e 1 ;:::;e k } of L such that e 1 = X. Define the Ricci curvature Ric L of L at X by (1.7) Ric L.X/ = K 12 + K K 1k ; where K ij denotes the sectional curvature of the 2-plane section spanned by e i, e j.we simply called such a curvature a k-ricci curvature. The scalar curvature of the k-plane section L is given by (1.8).L/ = K ij : 1 i< j k For each integer k, 2 k n, theriemannian invariant 2 k on an n-dimensional Riemannian manifold M is defined by (1.9) 2 k.p/ = 1 k 1 inf Ric L.X/; p M; L;X where L runs over all k-plane sections in T p M and X runs over all unit vectors in L. Recall that for a submanifold M in a Riemannian manifold, the relative null space of M at a point p M is defined by (1.10) N p ={X T p M h.x; Y / = 0; for all Y T p M}: 2. Ricci curvature and squared mean curvature Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for submanifolds in real space forms (see []). We prove similar inequalities for certain submanifolds of a Sasakian space form. A submanifold M normal to ¾ in a Sasakian space form M.c/ is called a C-totally real submanifold. It follows that maps any tangent space of M into the normal space, that is.t p M/ T p M, for every p M. THEOREM 2.1. Let M be an n-dimensional C-totally real submanifold of a./- dimensional Sasakian space form M.c/. Then:.i/ For each unit vector X T p M, we have (2.1) Ric.X/ 1 {.n 1/.c + 3/ + n2 H 2 }:.ii/ If H.p/ = 0, then a unit tangent vector X at p satisfies the equality case of (2.1) if and only if X N p.

4 250 Ion Mihai [].iii/ The equality case of (2.1) holds identically for all unit tangent vectors at p if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point. PROOF. (i) Let X T p M be a unit tangent vector X at p. We choose an orthonormal basis e 1 ;:::;e n ; e n+1 ;:::;e = ¾, such that e 1 ;:::;e n are tangent to M at p, with e 1 = X. Then, from the equation of Gauss, we have (2.2) n 2 H 2 = 2 + h 2 n.n 1/.c + 3/=: From (2.2), we get (2.3) n 2 H 2 = = m r=n+1 2m 2m [.h r 11 /2 +.h r hr nn /2 + 2 ].h r ij /2 i< j r=n+1 2m r=n+1 i< j h r ii hr jj n.n 1/c + 3 [.h r hr nn /2 +.h r 11 hr 22 hr nn /2].h r ij/ 2 2 From the equation of Gauss, we find (2.) and consequently (2.5) K ij = 2 i< j n K ij = 2m 2m r=n+1 Substituting (2.5)in(2.3), one gets 2m [h r ii hr jj.hr ij /2 ]+ c + 3 ; [h r ii hr jj.hr ij /2 ]+ h r iih r jj n.n 1/ c + 3 :.n 1/.n 2/ c : (2.6) n 2 H n2 2 H m r=n+1 j=2.h r 1 j/ 2 2 K ij 2.n 1/ c + 3 : 2 i< j n Therefore, n 2 H 2 =2 2Ric.X/ 2.n 1/.c + 3/= or equivalently (2.1). (ii) Assume H. p/ = 0. Equality holds in (2.1) if and only if { h r (2.7) 12 = = hr 1n = 0; h r 11 = hr hr nn ; r {n + 1;:::;2m}:

5 [5] Ricci curvature of submanifolds 251 Then h r 1 j = 0, for every j {1;:::;n}, r {n + 1;:::;2m}, that is X N p. (iii) The equality case of (2.1) holds for all unit tangent vectors at p if and only if (2.8) { h r ij = 0; h r hr nn 2hr ii = 0; i = j; r {n + 1;:::;2m}; i {1;:::;n}; r {n + 1;:::;2m}: We distinguish two cases.a/ n = 2, then p is a totally geodesic point;.b/ n = 2, it follows that p is a totally umbilical point. The converse is trivial. In the following we will consider submanifolds M tangent to the Reeb vector field ¾. THEOREM 2.2. Let M.c/ be a.2m + 1/-dimensional Sasakian space form and M an n-dimensional submanifold tangent to ¾. Then:.i/ For each unit vector X T p M orthogonal to ¾, we have (2.9) Ric.X/ {.n 1/.c + 3/ +.3 PX 2 2/.c 1/=2 + n 2 H 2} =:.ii/ If H.p/ = 0, then a unit tangent vector X T p M orthogonal to ¾ satisfies the equality case of (2.9) if and only if X N p..iii/ The equality case of (2.9) holds identically for all unit tangent vectorsorthogonal to ¾ at p if and only if p is a totally geodesic point. PROOF. Let X T p M be a unit tangent vector X at p, orthogonal to ¾. We choose an orthonormal basis e 1 ;:::;e n = ¾;e n+1 ;:::;e such that e 1 ;:::;e n are tangent to M at p, with e 1 = X. Then, from the equation of Gauss, we have (2.10) n 2 H 2 = 2 + h 2 n.n 1/.c + 3/=.3 P 2 2n + 2/.c 1/=: From (2.10), we get (2.11) n 2 H 2 = 2 + r=n+1 2 = [.h r 11 /2 +.h r hr nn /2 + 2 ].h r ij /2 i< j r=n+1 h r ii hr jj n.n 1/c+3.3 P 2 2n +2/ c 1 [.h r hr nn /2 +.h r 11 hr 22 hr nn /2]

6 252 Ion Mihai [6] + 2 r=n+1 i< j.h r ij /2 2.3 P 2 2n + 2/ c 1 : From the equation of Gauss, we find (2.12) 2 i< j n K ij = [h r ii hr jj.hr ij /2 ]+ h r ii hr jj n.n 1/c + 3.n 1/.n 2/ c P 2 3 Pe 1 2 2n + / c 1 8 : Substituting (2.12)in(2.11), as in the proof of Theorem 2.1 one gets n 2 H 2 =2 2Ric.X/ 2.n 1/.c + 3/=.3 PX 2 2/.c 1/=; which is equivalent to (2.9). The proofs of (ii) and (iii) are similar to their corresponding statements of Theorem 2.1. In this case,since ¾ is tangent to M, it follows that a totally umbilical point is totally geodesic. A submanifold M tangent to ¾ is said to be invariant (respectively anti-invariant)if.t p M/ T p M, for every p M (respectively.t p M/ T p M, for every p M). COROLLARY 2.3. Let M be an n-dimensional invariant submanifold of a Sasakian space form M.c/. Then:.i/ For each unit vector X T p M orthogonal to ¾, we have (2.13) Ric.X/ {.n 1/.c + 3/ +.c 1/=2}=:.ii/ A unit tangent vector X T p M orthogonal to ¾ satisfies the equality case of (2.13) if and only if X N p..iii/ The equality case of (2.13) holds identically for all unit tangent vectors orthogonal to ¾ at p if and only if p is a totally geodesic point. COROLLARY 2.. Let M be an n-dimensional anti-invariant submanifold of a Sasakian space form M.c/. Then:.i/ For each unit vector X T p M orthogonal to ¾, we have (2.1) Ric.X/ {.n 1/.c + 3/.c 1/ + n 2 H 2 }=:.ii/ If H.p/ = 0, then a unit tangent vector X T p M orthogonal to ¾ satisfies the equality case of (2.1) if and only if X N p.

7 [7] Ricci curvature of submanifolds 253.iii/ The equality case of (2.1) holds identically for all unit tangent vectors orthogonal to ¾ at p if and only if p is a totally geodesic point. A submanifold M tangent to ¾ is called a contact CR-submanifold [8] if there exists a pair of orthogonal differentiable distributions D and D on M, such that:.i/ TM = D D {¾}, where{¾} is the 1-dimensional distribution spanned by ¾..ii/ D is invariant by, that is.d p / D p, for every p M..iii/ D is anti-invariant by, that is.d / T p p M, for every p M. COROLLARY 2.5. Let M be an n-dimensional contact CR-submanifold of a Sasakian space form M.c/. Then:.i/ For each unit vector X D p, we have Ric.X/ {.n 1/.c + 3/ +.c 1/=2 + n 2 H 2} =:.ii/ For each unit vector X D p, we have Ric.X/ {.n 1/.c + 3/.c 1/ + n 2 H 2} =: 3. k-ricci curvature In this section, we prove a relationship between the k-ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. First, we state an inequality between the scalar curvature and the squared mean curvature for C-totally real submanifolds. THEOREM 3.1. Let M be an n-dimensional C-totally real submanifold of a Sasakian space form M.c/. Then we have (3.1) H 2 2 n.n 1/ c + 3 : PROOF. We choose an orthonormal basis {e 1 ;:::;e n ; e n+1 ;:::;e = ¾} at p such that e n+1 is parallel to the mean curvature vector H.p/ and e 1 ;:::;e n diagonalize the shape operator A n+1. Then the shape operators take the forms a a 2 0 A n+1 = : : : :: : ; 0 0 a n

8 25 Ion Mihai [8] A r = ( hij) r ; i; j = 1;:::;n; r = n + 2;:::;2m; trace Ar = h r ii = 0: From (2.2), we get (3.2) n 2 H 2 = 2 + a 2 i + 2m r=n+2 i; j=1.h r ij /2 n.n 1/ c + 3 : On the other hand, since 0.a i< j i a j / 2 =.n 1/ i a2 i 2 a i< j ia j,we obtain ( ) 2 n 2 H 2 = a i = a 2 i + 2 (3.3) a i a j n a 2 i ; i< j which implies n a2 i n H 2. We have from (3.2) (3.) n 2 H n H 2 n.n 1/.c + 3/=; which is equivalent to (3.1). Using Theorem 3.1, we obtain the following. THEOREM 3.2. Let M be an n-dimensional C-totally real submanifold M of a Sasakian space form M.c/. Then, for any integer k, 2 k n, and any point p M, we have (3.5) H 2.p/ 2 k.p/.c + 3/=: PROOF. Let {e 1 ;:::;e n } be an orthonormal basis of T p M. Denote by L i1:::i k the k-plane section spanned by e i1 ;:::;e ik. It follows from (1.7) and (1.8) that (3.6).L i1 i k / = 1 2 i {i 1;:::;i k} Ric Li1 i k.e i / and.p/ = 1 C k 2 n 2 1 i 1< <i k n.l i1 i k /: Combining (1.9) and (3.6), we find that.p/ n.n 1/2 k.p/=2, which together with (3.1) gives us (3.5). Next, we obtain analogous estimates for submanifolds tangent to ¾. THEOREM 3.3. Let M.c/ be a Sasakian space form and M an n-dimensional submanifold tangent to ¾. Then we have (3.7) H 2 2 n.n 1/ c P 2 2n + 2/.c 1/ : n.n 1/

9 [9] Ricci curvature of submanifolds 255 PROOF. We choose an orthonormal basis {e 1 ;:::;e n = ¾;e n+1 ;:::;e } at p such that e n+1 is parallel to the mean curvature vector H.p/ and e 1 ;:::;e n diagonalize the shape operator A n+1. Then the shape operators take the forms a a 2 0 A n+1 = : : : :: : ; 0 0 a n A r =.h r ij /; i; j = 1;:::;n; r = n + 2;:::;2m + 1; trace A r = From (2.10), we get n 2 H 2 = 2 + a 2 i + r=n+2 i; j=1.3 P 2 2n + 2/.c 1/=: Since, by (3.), we have n a2 i n H 2, it follows that n 2 H n H 2 n.n 1/ c + 3 which is equivalent to (3.7). From (3.6) and (3.7), we obtain the following theorem. h r ii = 0:.h r ij /2 n.n 1/.c + 3/=.3 P 2 2n + 2/ c 1 ; THEOREM 3.. Let M.c/ be a Sasakian space form and M an n-dimensional submanifold tangent to ¾. Then, for any integer k, 2 k n, and any point p M,we have (3.8) H 2.p/ 2 k.p/ c P 2 2n + 2/.c 1/ : n.n 1/ COROLLARY 3.5. Let M be an n-dimensional invariant submanifold of a Sasakian space form M.c/. Then, for any integer k, 2 k n, and any point p M, we have 2 k.p/.c + 3/= +.c 1/=.n/. COROLLARY 3.6. Let M be an n-dimensional anti-invariant submanifold of a Sasakian space form M.c/. Then, for any integer k, 2 k n, and any point p M, we have H 2.p/ 2 k.p/.c + 3/= +.c 1/=.2n/.

10 256 Ion Mihai [10] COROLLARY 3.7. Let M be an n-dimensional contact CR-submanifold of a Sasakian space form M.c/. Then, for any integer k, 2 k n, and any point p M,we have where 2h = dim D. H 2.p/ 2 k.p/ c + 3.3h n + 1/.c 1/ ; 2n.n 1/ References [1] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509 (Springer, Berlin, 1976). [2] B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. 60 (1993), [3], Mean curvature and shape operator of isometric immersions in real space form, Glasgow Math. J. 38 (1996), [], Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. 1 (1999), [5] F. Defever, I. Mihai and L. Verstraelen, B. Y. Chen s inequality for C-totally real submanifolds of Sasakian space forms, Boll. Un. Mat. Ital. (7) 11 (1997), [6] K. Matsumoto, I. Mihai and A. Oiagă, Ricci curvature of submanifolds in complex space forms, Rev. Roum. Math. Pures Appl. (to appear). [7] I. Mihai, R. Rosca and L. Verstraelen, Some aspects of the differential geometry of vector fields, Vol. 2 (PADGE, K. U. Leuven, K. U. Brussels, 1996). [8] K. Yano and M. Kon, Structures on manifolds (World Scientific, Singapore, 198). Faculty of Mathematics University of Bucharest Str. Academiei Bucharest Romania imihai@math.math.unibuc.ro