Totally Umbilical Space-like Submanifolds In Space Forms And Curvature And Topology

In this thesis, we mainly study the totally umbilical space-like submani-folds in space forms. Besides,we study the topology of Riemannian manifolds with some curvature conditions via comparisonsl geometry methods,and we get some results towards the topological type and the finiteness of isomorphism classes of fundamental groups of some Riemannian manifolds.In chapter one, we introduce the standard models of space forms.In chapter two, we prove that connected compact maximal space-like sub-manifolds in the space formS_p^m+1(1) (?) E_p^m+p+1 (p ≥ 1) must be totally umbilical, and also totally geodesic. Particularly, when p = 1, our result is Montiel's in case of H = 0.In chapter three, we develop some integral formulas for compact space-like hypersurfaces in anti-de Sitter spacetime H_<sup>n+1 and apply them in order to characterize the totally umbilical ones in H_<sup>n+1 as the only compact space-like hypersurfaces with constant higher order mean curvature under some appropriate hypothesis.In chapter four, we prove that a complete open Riemmannian manifold with Ricci curvature negatively lower bounded is of finite topological type provided that the conjugate radius is bounded from below by a positive constant and its excess is bounded by some function of its conjugate radius. Besides, we prove that there are only finitely many isomorphism classes of fundamental groups in the class of compact Riemannian manifolds with Ricci curvature Ric(M) ≥ (n—1)k, semi-simple connected radius R(M) ≥ R₀ and diameter d(M) ≤ D. We also show that the order of the group generated by short loops must be finite in this class of manifolds.