Odd Wei Liman Manifold On A Rigid Phenomenon

This thesis consists of four sections.The first and second sections are introduction and preliminaries respectively.In the third section, We suppose that M2"+1 be a (2n+1)-dimensional compact, simply connected Riemannian manifold without boundary and S2"+1 be the unit sphere in Euclidean space R2n+2,According to the results we have, there exists a positive numberδ∈(0.117,0.25), such that any compact simply connected Riemannian manifold Mn with Inj(M)>π. Therefore,we prove 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 for some positive numberηdepending only on n, then M2n+1 is diffeomorphic to S2n+1, and according to the results,we finally prove that it's true for any dimension under the above volume condition.In the fourth section,we prove that the injectivity radius have the rigidity phenomena. Under the assumption that Mn be a n-dimensional compact, simply connected Riemannian manifold without boundary and S" be the unit sphere in Euclidean space R"+1,and K is a positive number, We prove in this section whenever the manifold concerned satisfies that the sectional curvature Km varies in and the injectivity radius Inj(M) varies in [π-η,π],for some positive numberηdepending only on n, and joint with the above the-orems,under the above rigidity theorems of the injectivity radius, we finally prove a theorem: when the volume condition changes to ,we also have Mn is diffeomor-phic to S".