Some Results On Curvature And Topology Of Open Manifold

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 Rⁿ. 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 n-manifold satisfyingSuppose thatthen (M,g) has finite topology type.2. Given positive numbersα> 0, r₀ > 0 and an integer n≥2, there is an∈=∈(n,α,r₀) > 0, such that any complete Riemannian n-manifold M,with Ricci curvature Ric_M≥0,α_m≥α, K_p^min≥-1, c_p≥r₀ andfor some p∈Mand allr≥r₀ is diffeomorphic to Rⁿ.3. Givenα∈((1/2),1)and an integer n≥2,there is constant r₀ = r₀(α,n),∈=∈(n,α)> 0, such that any complete Riemannian n-manifold M, with Ric_M≥0,α_M≥α,K_M^min≥-1 andfor some p∈M and all r≥r₀ is diffeomorphic to Rⁿ. 4. Let M be a complete n-manifold, K_M^min≥0 ifthen M is diffeomorphic to Rⁿ.