By BAI~G-YEN CHEN 1. Introduction. Let N be a submanifold of a Riemannian manifold M with g as its first fundamental form. Let V and V be the covariant differentiations on iV and M, respectively. The second fundamental form h of the immersion is given by (1.1) h(X, r) = vxr - vxr where X and Y are vector fields tangent to N. It is well-known that h is a normal- bundle-valued symmetric 2-form on N. If the first and second fundamental forms are proportional, there is a normal vector field H, called the mean-curvature vector, on N such that (1.2) h(X, r) -~ g(X, Y)H. In this case, N is called a totally umbilical submani/old of M. The snbmanifold h r is called a totally geodesic submani/old ff h vanishes identically. Totally umbilical submanifolds (including totally geodesic submanifolds) are the simplest submani- folds of Riemannian manifolds. In this paper we shall study totally umbilical sub- manifolds in Kaehler manifolds. In particular we shall prove the following Main Theorem. Let M be a Hermitian symmetric space and N a totally umbilical hypersur/ace o/M. (A) I] N is totally geodesic, then M is the Riemannian product o/ a Hermitian sym- metric space M1 and a two-dimensional Hermitian symmetric space Ms such that !Y is locally the Riemannian product o/M1 and a geodesic o/Ms. (B) I] N is not totally geodesic, then one o/the ]ollowing three statements holds: (a) M is a complex projective line C,P 1 (c), a complex line C 1, or a disk D1 (--c) in G 1 and N any curve in M. Here c and --c denote the sectional curvature o/C,P 1 and D 1, reslaectively. (b) N is a totally umbilical hypersur/ace o/M and M is the Riemannian product CP 1 (c) X D 1 (-- c). (c) M is the complex number m-space C m and N is an open portion o/an ordinary hypersphere o/G m. I/statement (b) holds, then, locally, N is the con/ormal image o/a hypersphere S a or a hyperplane Ea o/C ~ under a con/ormal mapping/tom C ~ into M. 6*