RICCI TENSOR OF SLANT SUBMANIFOLDS IN COMPLEX SPACE FORMS. Abstract

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1 K. MATSUMOTO, I. MIHAI AND Y. TAZAWA KODAI MATH. J. 26 (2003), RICCI TENSOR OF SLANT SUBMANIFOLDS IN COMPLE SPACE FORMS Koji Matsumoto, Ion Mihai* and Yoshihiko Tazawa Abstract B.-Y. 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. The Lagrangian version of this inequality was proved by the same author. In this article, we obtain a sharp estimate of the Ricci tensor of a slant submanifold M in a complex space form ~Mð4cÞ, in terms of the main extrinsic invariant, namely the squared mean curvature. If, in particular, M is a Kaehlerian slant submanifold which satisfies the equality case identically, then it is minimal. 1. Preliminaries Let M be a real n-dimensional submanifold of a complex m-dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. We denote by and ~ the Levi-Civita connections of M and ~Mð4cÞ, respectively. Let J be the complex structure on ~Mð4cÞ. Also, we denote by h the second fundamental form and R the Riemann curvature tensor of M. Then the Gauss equation is given by ð1:1þ ~Rð; Y; Z; WÞ ¼Rð; Y; Z; WÞ þ gðhð; WÞ; hðy; ZÞÞ gðhð; ZÞ; hðy; WÞÞ for any vectors ; Y; Z; W tangent to M, where ð1:2þ ~Rð; Y; Z; WÞ ¼cfgð; ZÞgðY; WÞ gð; WÞgðY; ZÞ gðj; WÞgðJY; ZÞþgðJ; ZÞgðJY; WÞ þ 2gð; JYÞgðZ; JWÞg: 2000 Mathematics Subject Classification: 53C40, 53C25. Keywords: Ricci tensor, Ricci curvature, mean curvature, complex space form, Kaehlerian slant submanifold, totally real submanifold. * The second author was supported by a JSPS research fellowship. Received April 25, 2002; revised August 30,

2 86 koji matsumoto, ion mihai and yoshihiko tazawa Let p A M and fe 1 ;...; e 2m g an orthonormal basis at p, such that e 1 ;...; e n are tangent to M and e nþ1 ;...; e 2m are normal to M. We denote by H the mean curvature vector, i.e., ð1:3þ Also, we set HðpÞ ¼ 1 n hðe i ; e i Þ: ð1:4þ and h r ij ¼ gðhðe i; e j Þ; e r Þ; i; j A f1;...; ng; r A fn þ 1;...; 2mg ð1:5þ khk 2 ¼ n i; j¼1 gðhðe i ; e j Þ; hðe i ; e j ÞÞ: For any p A M and A T p M, we put J ¼ P þ F, where P and F are the tangential and normal components of J, respectively. We denote by ð1:6þ kpk 2 ¼ n i; j¼1 g 2 ðpe i ; e j Þ: We recall that for a submanifold M in a Riemannian manifold, the relative null space of M at a point p A M is defined by N p ¼f A T p M j hð; YÞ ¼0; for all Y A T p Mg: 2. Ricci tensor and squared mean curvature B.-Y. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for submanifolds in real space forms (see [3]). Afterwards, he obtained the Lagrangian version of this relationship (see [4]). First, we prove a similar inequality for an n-dimensional slant submanifold M of an m-dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. A submanifold M of a complex space form ~Mð4cÞ is said to be a slant submanifold [1] if for any p A M and any nonzero vector A T p M, the angle between J and the tangent space T p M is constant ð¼ yþ. It is obvious that both complex submanifolds and totally real submanifolds are slant submanifolds, corresponding to y ¼ 0 and y ¼ p=2, respectively. Theorem 2.1. Let M be an n-dimensional y-slant submanifold in an m- dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. Then:

3 ricci tensor of slant submanifolds 87 i) For each unit vector A T p M, we have ð2:1þ RicðÞ a n2 4 khk2 þðn 1Þc þ 3c cos 2 y: ii) If HðpÞ ¼0, then a unit tangent vector at p satisfies the equality case of (2.1) if and only if A N p. 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 ¼ 2 and p is a totally umbilical point. In the proof of this theorem, we will use the following result of B.-Y. Chen. ð2:2þ Lemma [2]. Let n b 2 and a 1 ;...; a n ; b real numbers such that a i! 2 ¼ðn 1Þ a 2 i þ b! : Then 2a 1 a 2 b b, with equality holding if and only if a 1 þ a 2 ¼ a 3 ¼¼a n : We will give a very short proof, di erent from the original one in [2]. Proof. By the Cauchy-Schwartz inequality, we have ½ða 1 þ a 2 Þþa 3 þþa n Š 2 a ðn 1Þ½ða 1 þ a 2 Þ 2 þ a 2 3 þþa2 n Š: The equation (2.2) implies or equivalently, 2a 1 a 2 b b. The equality holds if and only if a 2 i þ b a ða 1 þ a 2 Þ 2 þ a 2 3 þþa2 n a 1 þ a 2 ¼ a 3 ¼¼a n : r Proof of Theorem 2.1. i) Let A T p M be a unit tangent vector at p. We choose an orthonormal basis fe 1 ;...; e n ; e nþ1 ;...; e 2m g such that e 1 ;...; e n are tangent to M at p, with e n ¼ and e nþ1 is parallel to the mean curvature vector HðpÞ. Then, from the Gauss equation, we have ð2:3þ n 2 khk 2 ¼ 2t þkhk 2 ½nðn 1Þþ3n cos 2 yšc;

4 88 koji matsumoto, ion mihai and yoshihiko tazawa where t denotes the scalar curvature at p, thatis, ð2:4þ We put t ¼ 1ai< jan Kðe i 5e j Þ¼ 1ai< jan Rðe i ; e j ; e i ; e j Þ: d ¼ 2t n2 2 khk2 ½nðn 1Þþ3n cos 2 yšc: Then, from (2.3), we get n 2 khk 2 ¼ 2ðd þkhk 2 Þ: With respect to the above orthonormal basis, (2.4) takes the following form:! 2 ( hii nþ1 ¼ 2 d þ n ðhii nþ1 Þ 2 þ ) ðhij nþ1 Þ 2 þ 2m ðhij r Þ2 : i0j r¼nþ2 i; j¼1 If we put a 1 ¼ h11 nþ1, a 2 ¼ P n 1 i¼2 hnþ1 ii and a 3 ¼ hnn nþ1, the above equation becomes! 3 2 ( a i ¼ 2 d þ 3 ða i Þ 2 þ ) ðhij nþ1 Þ 2 þ 2m ðhij r Þ2 haa nþ1 hnþ1 bb : i0j r¼nþ2 i; j¼1 2aa0ban 1 Thus a 1 ; a 2 ; a 3 satisfy the Lemma of Chen (for n ¼ 3), i.e., 3 a i! 2 ¼ 2 b þ 3 ða i Þ 2!: Then 2a 1 a 2 b b, with equality holding if and only if a 1 þ a 2 ¼ a 3. In the case under consideration, this means or equivalently, 1aa0ban 1 h nþ1 aa hnþ1 bb b d þ 2 i<j ðhij nþ1 Þ 2 þ 2m r¼nþ2 i; j¼1 ðh r ij Þ2 ð2:5þ n 2 2 khk2 þ½nðn 1Þþ3n cos 2 yšc b 2t 1aa0ban 1 haa nþ1 hnþ1 bb þ 2 i<j ðhij nþ1 Þ 2 þ 2m r¼nþ2 i; j¼1 ðh r ij Þ2 : Using again the Gauss equation, we have

5 ricci tensor of slant submanifolds 89 ð2:6þ 2t 1aa0ban 1 haa nþ1 hnþ1 bb þ 2 i<j ðhij nþ1 Þ 2 þ 2m r¼nþ2 i; j¼1 ðh r ij Þ2 ¼ 2Sðe n ; e n Þþ½ðn 1Þðn 2Þþ3ðn 2Þ cos 2 yšc þ 2 n 1 þ 2m r¼nþ2 8 < ðhnn r Þ2 þ 2 n 1 ðhin r Þ2 þ : where S is the Ricci tensor of M. Combining (2.5) and (2.6), we obtain! 2 n 1 haa r a¼1 9 = ; ; ðh nþ1 in Þ 2 n 2 2 khk2 þ½2ðn 1Þþ6 cos 2 yšc b 2Sðe n ; e n Þþ2 n 1 ðhin nþ1 Þ 2 þ 2m r¼nþ2 8 < : ðh r in Þ2 þ! 2 n 1 haa r a¼1 9 = ; which implies (2.1). ii) Assume Hð pþ ¼ 0. Equality holds in (2.1) if and only if ( h1n r ¼¼hr n 1; n ð2:7þ ¼ 0 hnn r ¼ P n 1 ; r A fn þ 1;...; 2mg: hr ii Then hin r ¼ 0, Ei A f1;...; ng, r A fn þ 1;...; 2mg, i.e., A N p. iii) The equality case of (2.1) holds for all unit tangent vectors at p if and only if hij r ¼ 0; i 0 j; r A fn þ 1;...; 2mg; ð2:8þ h11 r þþhr nn 2hr ii ¼ 0; i A f1;...; ng; r A fn þ 1;...; 2mg: We distinguish two cases: a) n 0 2, then p is a totally geodesic point; b) n ¼ 2, it follows that p is a totally umbilical point. The converse is trivial. r Corollary 2.2. Let M be an n-dimensional totally real submanifold in an m-dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. Then: i) For each unit vector A T p M, we have ð2:9þ RicðÞ a n2 4 khk2 þðn 1Þc: ii) If HðpÞ ¼0, then a unit tangent vector at p satisfies the equality case of (2.9) if and only if A N p.

6 90 koji matsumoto, ion mihai and yoshihiko tazawa iii) The equality case of (2.9) holds identically for all unit tangent vectors at p if and only if either p is a totally geodesic point or n ¼ 2 and p is a totally umbilical point. It is known that every complex submanifold of a Kaehlerian manifold is minimal. Corollary 2.3. Let M be an n-dimensional complex submanifold in an m- dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. Then: i) For each unit vector A T p M, we have ð2:10þ RicðÞ a 2ðn þ 1Þc: ii) A unit tangent vector at p satisfies the equality case of (2.10) if and only if A N p. iii) The equality case of (2.10) holds identically for all unit tangent vectors at p if and only if p is a totally geodesic point. By polarization, from Theorem 2.1, we derive: Theorem 2.4. Let M be an n-dimensional y-slant submanifold in an m- dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. Then the Ricci tensor S satisfies ð2:11þ S a n2 4 khk2 þðn 1Þcþ3c cos 2 y g: The equality case of (2.11) holds identically if and only if either M is a totally geodesic submanifold or n ¼ 2 and M is a totally umbilical submanifold. In particular, for totally real and complex submanifolds, respectively, we state: Corollary 2.5 [4]. Let M be an n-dimensional totally real submanifold in an m-dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. Then the Ricci tensor S satisfies ð2:12þ S a n2 4 khk2 þðn 1Þc g: The equality case of (2.12) holds identically if and only if either M is a totally geodesic submanifold or n ¼ 2 and M is a totally umbilical submanifold. For a classification of totally umbilical submanifolds in nonflat complex space forms we refer to [6].

7 Corollary 2.6. Let M be an n-dimensional complex submanifold in an m- dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. Then the Ricci tensor S satisfies ð2:13þ ricci tensor of slant submanifolds 91 S a 2ðn þ 1Þcg: The equality case of (2.13) holds identically if and only if M is a totally geodesic submanifold. 3. Minimality of Kaehlerian slant submanifolds Let ~Mð4cÞ be an n-dimensional complex space form of constant holomorphic sectional curvature 4c and M an n-dimensional y-slant submanifold of ~Mð4cÞ. By reference to [1], M is said to be a Kaehlerian slant submanifold if it is proper (i.e., y B f0; p=2g) and the endomorphism P of the tangent bundle TM is parallel with respect to the Riemannian connection of M (i.e. P ¼ 0). A Kaehlerian slant submanifold is a Kaehler manifold with respect to the induced metric and the almost complex structure ~J ¼ð1=cos yþp. It is known that every proper slant surface in a Kaehler manifold is Kaehlerian slant (see [1]). An example of a 4-dimensional Kaehlerian slant submanifold in C 4 is given by the following immersion. xðu; v; w; zþ ¼ðu; v; k sin w; k sin z; kw; kz; k cos w; k cos zþ; where k > 0 is a constant. In this case, y ¼ p=4 (see [1]). We denote by R the maximum Ricci curvature function on M (see [4]), defined by RðpÞ ¼maxfSðu; uþju A Tp 1 Mg; p A M; where Tp 1M ¼fu A T pm j gðu; uþ ¼1g. If n ¼ 3, R is the Chen first invariant d M defined in [2]. For n > 3, R is the Chen invariant dðn 1Þ (see [5]). In this section, we derive an inequality for the Chen invariant R and prove that any Kaehlerian slant submanifold which satisfies the equality case is minimal. This is a generalization of a result of B.-Y. Chen [4] for Lagrangian submanifolds in complex space forms. Theorem 3.1. Let M be an n-dimensional Kaehlerian slant submanifold in an n-dimensional complex space form ~Mð4cÞ of constant holomorphic sectional curvature 4c. Then ð3:1þ R a n2 4 khk2 þðn 1Þcþ3c cos 2 y: If M satisfies the equality case of (3.1) identically, then M is a minimal submanifold.

8 92 koji matsumoto, ion mihai and yoshihiko tazawa Proof. The inequality (3.1) is an immediate consequence of the inequality (2.11). We assume that M is a Kaehlerian slant submanifold of ~Mð4cÞ, which satisfies the equality case of (3.1) at a point p A M. We may choose an orthonormal basis fe 1 ;...; e n g of T p M such that RðpÞ ¼Sðe n ; e n Þ. We set e nþj ¼ð1=sin yþfe j, j A f1;...; ng. By the proof of Theorem 2.1, it follows that the equations (2.7) hold, where hij r are the coe cients of the second fundamental form with respect to the orthonormal basis fe 1 ;...; e n ; e nþ1 ;...; e 2n g. Let A denote the shape operator of M in ~Mð4cÞ. It is known (see [1]) that P is parallel if and only if ð3:2þ A F Y ¼ A FY ; for all vector fields ; Y tangent to M. We distinguish two cases: i) If gðhðu; vþ; FwÞ ¼0, Eu; v; w A T p M, then obviously HðpÞ ¼0. ii) We assume that case i) does not hold. Then we define f p : T 1 p M! R; f pðvþ ¼gðhðv; vþ; FvÞ: Since Tp 1 1 M is compact, there exists a vector v A Tp M such that f p attains an absolute maximum at v. Let denote e 1 ¼ v and f p ðvþ ¼l 1 > 0. It follows that A Fe1 e 1 ¼ l 1 e 1. We can choose an orthonormal basis fe 1 ;...; e n g of T p M such that e i is an eigenvector of A Fe1 with corresponding eigenvalue l i, for all i A f1;...; ng. We consider the function f i ðtþ ¼f p ððcos tþe 1 þðsin tþe i Þ, i A f2;...; ng. It is easily seen that f i has a relative maximum at t ¼ 0. Thus, fi 0 ð0þ ¼0 and fi 00 ð0þ a 0. By a straightforward computation, one finds 0 b fi 00 ð0þ ¼ 3l 1 þ 6l i ; i.e., l 1 b 2l i, Ei b 2. Since l 1 > 0, one gets l 1 0 l i, Ei b 2. Thus, the multiplicity of the eigenvalue l 1 is 1. We have e 1 0Ge n. Otherwise A Fei e n ¼ GA Fei e 1 ¼ GA Fe1 e i ¼ Gl i e i? e n ; i A f2;...; ng; implies l 2 ¼¼l n ¼ 0, and hence, using (2.7), l 1 ¼ 0, which is a contradiction. On the other hand, by (2.7) it is easily seen that e n is an eigenvector of A Fe1. Thus, we can choose e n ¼ e n, and, consequently, we may assume e j ¼ e j, Ej A f1;...; ng. By (3.2) and (2.7), we have A Fen e 1 ¼ A Fe1 e n ¼ l n e n ¼ 0: Thus, (2.7) implies l 1 þþl n 1 ¼ l n ¼ 0. Therefore tr A Fe1 ¼ 0. For i A f2;...; n 1g, one has

9 tr A Fei ¼ n gða Fei e j ; e j Þ¼ n gðhðe j ; e j Þ; Fe i Þ¼2gðhðe n ; e n Þ; Fe i Þ j¼1 j¼1 ¼ 2gðhðe i ; e n Þ; Fe n Þ¼0: Similarly tr A Fen ¼ n j¼1 ricci tensor of slant submanifolds 93 gðhðe j ; e j Þ; Fe n Þ¼2 n 1 gðhðe j ; e j Þ; Fe n Þ¼2 n 1 gðhðe j ; e n Þ; Fe j Þ¼0: j¼1 Thus, tr A Fei ¼ 0, Ei A f1;...; ng. Consequently, HðpÞ ¼0. Corollary 3.2. Let M be an n-dimensional Kaeherian slant submanifold of an n-dimensional complex space form ~Mð4cÞ. If dim N p is positive constant, then M satisfies the equality case of (3.1) identically and is foliated by totally geodesic submanifolds. Proof. By the above proof, it follows that M satisfies the equality case of (3.1) at a point p A M if and only if dim N p b 1. Assume that dim N p is positive constant. It is known that N is involutive and its leaves are totally geodesic (see, for instance, [4], [10]). This achieves the proof. r j¼1 r Acknowledgements. valuable comments. The authors would like to thank the referee for his References [ 1 ] B.-Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Louvain, [ 2 ] B.-Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel), 60 (1993), [ 3 ] B.-Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasg. Math. J., 41 (1999), [ 4 ] B.-Y. Chen, On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms, Arch. Math. (Basel), 74 (2000), [ 5 ] B.-Y. Chen, Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. (N.S.), 26 (2000), [ 6 ] B.-Y. Chen and K. Ogiue, Two theorems on Kaehler manifolds, Michigan Math. J., 21 (1974), [ 7 ] B.-Y. Chen and Y. Tazawa, Slant submanifolds of complex projective and complex hyperbolic spaces, Glasg. Math. J., 42 (2000), [ 8] K. Matsumoto, I. Mihai and A. Oiagă, Ricci curvature of submanifolds in complex space forms, Rev. Roumaine Math. Pures Appl., 46 (2001), [9] I. Mihai, R. Rosca and L. Verstraelen, Some Aspects of the Di erential Geometry of Vector Fields, Centre Pure Appl. Di erential Geom. (PADGE) 2, Katholieke Universiteit Brussel, Brussels, Katholieke Universiteit Leuven, Louvain, 1996.

10 94 koji matsumoto, ion mihai and yoshihiko tazawa [10] H. Reckziegel, On the eigenvalues of the shape operator of an isometric immersion into a space of constant curvature, Math. Ann., 243 (1979), Department of Mathematics Faculty of Education Yamagata University Yamagata Japan ej192@kdw.kj.yamagata-u.ac.jp Faculty of Mathematics University of Bucharest Str. Academiei Bucharest Romania imihai@math.math.unibuc.ro School of Information Environment Tokyo Denki University Inzai Chiba Pref Japan tazawa@cck.dendai.ac.jp