Some Results on Curvature and Topology of Open Manifold 

Author  LuWen 
Tutor  LiGongBao 
School  Central China Normal University 
Course  Basic mathematics 
Keywords  large volume growth finite topological type diffeomophic 
CLC  O189.31 
Type  Master's thesis 
Year  2007 
Downloads  9 
Quotes  0 
In this paper, we study the noncompact manifold by virtue of comparison, more precisely, conditions on curvature are given to ensure the noncompact manifold being diffeomorphic to R^{n}. As we know, the study of noncompact manifolds turns more complicated than the compact ones,whose attributes guarantee simpler calculations and estimates.Here as follows are our main results.1. Let （M,g）be a complete nmanifold satisfyingSuppose thatthen （M,g） has finite topology type.2. Given positive numbersα> 0, r_{0} > 0 and an integer n≥2, there is an∈=∈（n,α,r_{0}） > 0, such that any complete Riemannian nmanifold M,with Ricci curvature Ric_{M}≥0,α_{m}≥α, K_{p}^{min}≥1, c_{p}≥r_{0} andfor some p∈Mand allr≥r_{0} is diffeomorphic to R^{n}.3. Givenα∈（（1/2）,1）and an integer n≥2,there is constant r_{0} = r_{0}（α,n）,∈=∈（n,α）> 0, such that any complete Riemannian nmanifold M, with Ric_{M}≥0,α_{M}≥α,K_{M}^{min}≥1 andfor some p∈M and all r≥r_{0} is diffeomorphic to R^{n}. 4. Let M be a complete nmanifold, K_{M}^{min}≥0 ifthen M is diffeomorphic to R^{n}.