TOTALLY REAL SUBMANIFOLDS IN A COMPLEX PROJECTIVE SPACE

In this paper, we establish the following result: Let M be an n-dimensional complete totally real minimal submanifold immersed in CPn with Ricci curvature bounded from below. Then either M is totally geodesic or infr ≤ (3n+1)(n−2)/3, where r is the scalar curvature of M .


Introduction.
Let CP n be the n-dimensional complex projective space with the Fubini-Study metric of constant holomorphic sectional curvature c = 4 and let M be an n-dimensional totally real submanifold of CP n .Let r be the scalar curvature of M. If M is compact, then many authors studied them and obtained many beautiful results (for example [2,4,5]).
In this paper, we make use of Yau's maximum principle to study the complete totally real minimal submanifold with Ricci curvature bounded from below and obtain the following result.
Theorem 1.Let M be an n-dimensional complete totally real minimal manifold immersed in CP n with Ricci curvature bounded from below.Then either M is totally geodesic or inf r ≤ (3n + 1)(n − 2)/3.

Preliminaries.
Let M be an n-dimensional totally real minimal submanifold of CP n .We choose a local field of orthonormal frames e 1 ,...,e n ,e 1 * = Je 1 ,...,e n * = Je n (J is the complex structure of CP n ), such that, restricted to M, the vectors e 1 ,...,e n are tangent to M. We make use of the following convention on the range of indices A,B,C,... = 1,...,n,1 * ,...,n * ; i,j,k,... = 1,...,n. (2.1) With respect to the frame field of CP n , let w A be the field of dual frames.Then the structure equations of CP n are given by where J = J AB e A ⊗ e B , so that where I n is the identity matrix of order n.We restrict these forms to M. Then from [2], we have ) ) ) (2.10) (2.12) The second fundamental form h of M in CP n is defined as If M is minimal in CP n , i.e., trace h = 0, then from (2.4) and (2.10), we have where r is the scalar curvature of M. Define h m * ijk and h m * ijkl by respectively.Let H l * and ∆ denote the (n×n)-matrix h l * ij and the Laplacian on M, respectively.By a simple calculation, we have (cf.[2]) (2.16) The following lemma is important in this paper.
Lemma 1 [6].Let M n be a complete Riemannian manifold with Ricci curvature bounded from below and let f be a C 2 -function bounded from above on M n , then for all > 0, there exists a point x ∈ M n at which