Research On The Curvature And The Topology Of Open Manifolds Of Volume Growth

The research on complete open Riemannian manifolds is an important topic in modern differential geometry,and the key point in this kind of problems is to understand the relationship between the curvature and the topology of complete open Riemannian manifolds.This paper mainly discusses how the curvature of manifolds determines their topological properties,namely,under certain conditions of volume growth and curvature,complete open Riemannian manifold has finite topological type,or a stronger result,is diffeomorphic to a Euclidean space.The specific research work is described as follows:1.In the aspect of large volume growth,this paper mainly studies the topological problems of complete open n-dimensional Riemannian manifolds with Ricci curvature bounded from constant below.Based on the Toponogov comparison theorem,the lower bound of Excess function can be estimated.Combining with the critical point theory,we prove that the manifold with radial sectional curvature bounded from below is diffeomorphic to a Euclidean space under certain large volume growth conditions.This result not only improves curvature and volume growth conditions,but also generalizes some conclusions about Riemannian manifolds with Ricci curvature bounded from constant below under large volume growth.2.In the aspect of sub-large volume growth,this paper mainly researches the topological problems of complete open n-dimensional Riemannian manifolds with nonnegative Ricci curvature.Based on the relationship between Busemann function and Excess function,the upper bound of Excess function can be estimated.Combining with the critical point theory,we prove that the manifold with lower bound of ?-sectional curvature decay has finite topological type under certain sub-large volume growth conditions.This consequence enriches the conclusions about Riemannian manifolds with lower bound of ?-sectional curvature decay under sub-large volume growth.3.In the aspect of weighted large volume growth,this paper mainly focuses on the topological problems of complete open n-dimensional Riemannian manifolds with Bakry-Emery Ricci curvature bounded from constant below.Based on the Toponogov comparison theorem,the upper bound of Excess function can be estimated.Combining with the critical point theory,we prove that the manifold with radial sectional curvature bounded from below has finite topological type under certain weighted large volume growth conditions.Some classical conclusions about Riemannian manifolds with nonnegative Ricci curvature under large volume growth are generalized.