SPACELIKE SUBMANIFOLDS IN INDEFINITE SPACE FORM Mn+p

Let Mn+p p (c) be n + p-dimensional connected semi-Riemannian manifold of constant curvature c whose index is p. It is called indefinite space form of index p. Let M be an n-dimensional Riemannian manifold immersed in Mn+p p (c). The semi-Riemannian metric of Mn+p p (c) induces the Riemannian metric of M , M is called a spacelike submanifold. Spacelike submanifolds in indefinite space form Mn+p p (c) have been of increasing interesting in the recent years. There are many results about these submanifolds, for instance, Dong [3], Wu [6, 7], Liu[4]. In [5], the authors got an intrinsic inequality for spacelike hypersurfaces in de Sitter space form Mn+1 1 whose index is 1. In this note, we generalize the intrinsic inequality for spacelike hypersurface of de Sitter space to spacelike submanifolds of indefinite space form Mn+p p (c) with index p ≥ 1. From this inequality, we also get some rigidity theorems for such spacelike submanifolds.