Totally real submanifolds of complex space forms, I

Introduction.Let M be a real 2m-dimensional Kaehler manifold with almost complex structure J.An ^-dimensional Riemannian manifold M isometrically immersed in M is called a totally real submanifold of M if T X (M) ±JT X (M) for each xeM, where T X (M) denotes the tangent space to M at x.Here we have identified T Z (M) with its image under the differential of the immersion because our computation is local.If X is a tangent vector of M at x, then JX is a normal vector to M. Thus we see that n ^ m.When n = m, totally real submanifolds have many interesting properties studied by different authors (see [1], [2], [4], [5], [6] and [8]).The purpose of the present paper is to study an ^-dimensional totally real submanifold M of a real 2^-dimensional complex space form M satisfying certain conditions on the second fundamental form of M. In §1 we state some fundamental formulas for totally real submanifolds of a Kaehler manifold.In §2 we prepare some lemmas which we need in the sequel.In §3 we prove the basic properties of an ^-dimensional totally real submanifold M of a real 2^-dimensional Kaehler manifold M with constant holomorphic sectional curvature under the condition that M has the parallel mean curvature vector.In the last section we prove our main theorems which give a characterization of a compact ^-dimensional flat totally real submanifold S^rJ x S 1 ^) x x S ι (r n ) in the flat Kaehler manifold C n of real dimension 2n. Preliminaries.Let M be a Kaehler manifold of real dimension 2n (complex dimension n) and M be an ^-dimensional totally real submanifold of M. We choose a local field of orthonormal frames e lf , e 2n in M in such a way that, restricted to M, the vectors e 19, e n are tangent to M, and hence remaining vectors e n+1 , •• ,β 2 » are normal to M. Unless stated otherwise, we use the following conventions that the ranges of indices are respectively:1 ^ A, B, C, ^ 2n , 1 ^ i, j, k, ^ n n + 1 ^ a, b t c, 5j 2n , PROOF.By the assumption and Proposition 6, we have c = 0 and the ambient space M is C n .Thus Theorem 2 follows from Theorem 1.