BANG-YEN CHEN, LIEVEN VANHECKE AND LEOPOLD VERSTRAELEN COMPLEX SUBMANIFOLDS OF SOME HERMITIAN SYMMETRIC SPACES 1. INTRODUCTION Let N be a complex submanifold in a Kaehler manifold M. Then the normal bundle v of N in M with the induced Hermitian structure forms a Hermifian vector bundle over N. We denote by qb(v) the restricted holonomy group of v. Let CP ~ (resp., H ~) be the n-dimensional* complex projective space (resp., Hermitian hyperbolic space) of constant holomorphic sectional curvature 1 (resp., - 1). In [2], it is proved that there exist no complex submanifolds in CP ~ and H ~ with trivial restricted holonomy group qblv). In this paper we prove the following. THEOREM. lf N is a complete complex submanifoM of the Hermitian sym- metric space CP ~ x Hm(n >>.m) with trivial restricted holonomy group dg(v), then N is either CP ~ or H m. Moreover the immersion of N in CP ~ × H m is given in the trivial way. 2. BASIC FORMULAS Let N be a p-dimensional complex submanifold in the Hermitian symmetric space M = CP ~ x H m. Let g, J, and V be the metric, complex structure and the connection on M. Then the curvature tensor R of Mis given by R(X, Y) = VxV r -- VreV x - Vtx, r j. Let El,. •., E2n+2m be an orthonormal frame on M, then the Ricci tensor S of M is given by 2n + 2m (2.1) s(x, r) = ~ R(eo, X; r, Co), ct=l where R(E,, X; Y, E~) = g(R(E~, X)Y, E,). Let V' be the induced connec- tion on the complex submanifold N. Then for all vector fields X, Ytangent to N. We have Vx r = V'~ r + h(X, Y), where h is the second fundamental form of N in M. Let ~ be a vector field normal to N, we write Vx~ = -A¢(X) + Dx~, • Dimensions of complex manifolds are complex dimensions. Geometriae Dedicata 7 (1978) 427-432. All Rights Reserved Copyright © 1978 by D. Reidel Publishing Company, Dordrecht, Holland