The Curvature And The Topology Of Open Manifolds And The Geometry Of Submanifolds

In this thesis, we mainly study the topology of a manifold and submanifolds of a manifold in the field of Riemannian Geometry.We discuss the relation between the curvature and the topology of some Riemannian manifolds in chapter one. We study complete noncompact open Riemannian manifolds with some conditions via comparisonal geometry methods. We prove that M with some bounded conditions has finite topological type or even diffeomorphic to the Euclid Space.In chapter two, submanifolds in the de Sitter space are considered. We prove that a compact submanifold in the de Sitter space is totally umbilical if the length square of the second fundamental form of M is bounded by some Pinching conditions.At last in chapter three, the topology of submanifolds in a sphere is discussed. We prove that M is homeomorphic to a sphere if the length of the second fundamental form of M is bounded by a constant.