Estimation Of The Diameter Of The Odd The Weili Man Manifolds And Positive Ricci Curvature Of Riemannian Manifolds

This dissertation consists of four sections.The first and second sections are introduction and preliminaries respectively.In the third section I studied an upper bound of the diameter of an odd-dimensional Rie-mannian manifold. Under the assumption that M2n+1 be a 2n+1-dimensional compact, simply connected Riemannian manifold without boundary, I derive an upper bound of the diameter dM<(?) in this note whenever the manifold concerned satisfies that the sectional curvature Km varies in [δ,1] and the volume V(M) is not larger than 2(1+η)V(B(?)) for some positive numberηdepending only on n, whereδ∈(0.117,0.25) is a constant as described in Theorem 2.11 and B(?) is the geodesic ball on S2n+1 with radius(?). Then the similar issues of the general dimensional manifold is studied and a gap phenomenon of the manifold concerned is given out. I finally give a lower bound of the first eigenvalueλ1 of Laplacian on manifold M,that is,λ1 bigger than by virtue of a gap phenomenon of the manifold.In the fourth section I introduce the concept of the k-th Ricci curvature on a point x of a manifold M. Under the assumption that Riemannian manifold satisfies that the sectional curva-ture and the volume for some positive numberηdepending only on n, I derive an upper bound of the diameter...