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Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.

This paper gives the intrinsic conditions for a compact space-like hypersurface in a de Sitter space to be isometric to a sphere.

Throughout this paper, by a Riemannian manifold we always mean an ndimensional connected and oriented manifold covered by a system of local coordinates {x1} (i=l, 2, ■■■ , n) and furnished with a Riemannian metric tensor g ―g^dx^dx1. We use the Einstein summation convention with respect to repeated indices. Furthermore, geometric objects and some functions appeared in this paper are always assumed to be smooth, unless otherwise stated. Let M be an n-dimensional Riemannian manifold with a metric tensor g. We use the standard notation for the covariant derivative 1, the exterior differential d, the codifferential d, the Laplacian A and the volume element dV of M. We denote by and | | the inner product and the norm induced in