-- TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 193, 1974 ON TOTALLY REAL SUBMANIFOLDS BANG-YEN C H E N ( ~ ) AND KOICHI OGIUE(2) ABSTRACT. Complex analytic submanifolds and totally real submanifolds are two typical classes among all submanifolds of an almost Hermitian mani- fold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained. 1. Introduction. Among all submanifolds of an almost Hermitian manifold, there are two typical classes: one is the class of holomorphic submanifolds and the other is the clasz of totally real submanifolds. A submanifold M of an almost Hermitian manifold M is called holomorphic (resp. totally real) if each tangent space of M is mapped into itself (resp. the normal space) by the almost complex 'V structure of M. There have been many results in the theory of holomorphic sub- manifolds; on the other hand, there have been only a few results in the theory of totally real submanifolds. The purpose of this paper is to study some fundamental properties of totally real submanifolds. In $2, we investigate the general properties of totally real submanifolds. In $$3 and 4 we consider totally real submanifolds of a complex space form. In the last section, we first give a characterization for totally real surfaces in a complex space form, later we prove some results for totally real or holomor- phic surfaces by using analytic function theory. 2. Totally real submanifolds. Let M be an n-dimensional Riemayian mani- 'V fold and M be a Kaehler manifold of dimension 2(n + p), P 2 0. Let ] be the 'V rCI almost complex structure of M and let g (resp.g) be the Riemannian metric of M Received by the editors April 6, 1973. AMS (MOS) subject classifications (1970). Primary 53B25, 53C40; Secondary 53A10, 53B35. Key words and phrases. Totally real submanifolds, minimal submanifolds, complex space forms, sectional curvature, scalar curvature. (1) Work done under partial support by NSF Grant No. 36684. (2) Work done under partial support by the Matsunaga Science Foundation. Copyright Q 1974. American Mathematical Society