Differential Sphere Theorem. Excess Pinching Riemannian Manifold

This dissertation consists of four sections.The first section is introduction.The second sections is preliminaries. We introduce some basic concepts of Riemannian manifold, difinitions and theorems appeared in the demonstration.In the third section, we introduce the concept of Gromov-Hausdorff distance, Gromov-Hausdorff convergence and harmonic coordinate. The geometrical element ofπ-excess is defined. After that we introduce even dimensional differentiable sphere theorem on Riemannian manifold. Some lemmas and propositions are given which are used in the demonstration.The fourth section includes the main theorem. At first, we prove an odd dimensional differ-entiable sphere theorem on Riemannian manifold with reverse excess pinching. Namely, under the assumption that M be a 2n+1-dimensional compact, simply connected Riemannian mani-fold without boundary with the metric g, there exists a constantδ∈(0.117,0.25) and a positive numberη, such thatδ≤KM≤1, eπ(M)≤η, then M is diffeomorphic to the standard sphere S". With the even dimensional theorem, we have a disoretional dimensional differentiable sphere theorem with reverse excess pinching. At last, we prove a rigidity phenomena on Riemannian manifold of injectivity radius.