Research On The Topology Of Riemannian Manifolds With Different Curvature And Volume Growth

In Riemannian geometry,much attention is paying to the relationship between curvature and topology.In this dissertation,we study the topology of manifolds under certain conditions of curvature and volume growth.Firstly,we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature.If M satisfies that large volume growth and sectional curvature bounded from nonnegative below,on the one hand,by using extensions of Excess function and Busemann function,we prove that they have finite topological type with asymptotically nonnegative kth-Ricci curvature.On the other hand,combining with the Topongov's comparison theorems of the sectional curvature and critical point theory,we prove that they have finite topological type.This result weakens the condition of the sectional curvature tok _p?r??-C/?1+r?^?,which complements the research results of Mahaman and Zhang.Secondly,we study the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature.If its functionf?r?=vol[B?p,r?]/I _n?r?r^n-1is monotone decreasing and critical radius has positive lower bound.By using Topongov's comparison theorems of the sectional curvature and the upper bound of the Excess function,and the critical point theory,we prove that the manifold is diffeomorphic to Rⁿ under k _p?r??-C/?1+r?^?and sub-large volume growth,where 0???2 and C>0.The result improves the speed of volume growth to the r^{[?1-?/2??k??n-2?]/?k+1?}.This consequence enriches these results proved by Zhan and Xue.Thirdly,we study the topology of complete noncompact Riemannian manifolds with Ricci curvature bounded from below by nonnegative constants.By using the estimates of Excess function,we obtain the uniform cut lemma with Ricci curvature bounded from below.Combining with radial density function and the halfway lemma,we prove that the manifold with linear diameter growth has a finitely generated fundamental group,which generalizes the results of Sormani to the case where the Ricci curvature bounded from below.