A Pinching Problem Of Submanifolds And The Topology Of Open Manifolds With Curvature Bounded From Below

In this thesis,we mainly study a Pinching problem of the square of the norm of the second fundamental form S on submanifolds in a locally symmetric Riemann manifold and the topology of complete open manifolds with curvature bounded from below, and we get some results.First we introduce the researching status about the Pinching problem of the square of the norm of the second fundamental form concisely, and give a survey of the researching history about the relation between the curvature and the topology of open manifolds. Then we obtain an integral inequality about the square of the norm of the second fundamental form S of submanifolds in a locally symmetric Riemann manifold and discuss its Pinching problem,which extend the conclusion of [6].At last in chapter three we consider the finite topological type problem of complete open manifolds under some condition bound and some curvature bound.We prove that for an open manifold M with curvature bounded from below,if its distance function is bounded,then M has finite topological type or even differential homeomorphism to R~n.