Riemannian Geometry And Non-smooth Analysis

In Riemannian geometry, the relationship between curvature and topology is one of the central issues. Among all the studies of this relationship, the sphere theorem is a typical one. After decades of development, Brendle and Schoen finally proved the differential sphere theorem by using the tool of Ricci flow.This paper is a report of [17] whose original purpose was to prove the differentiable sphere theorem by using the tool of non-smooth analysis. Unfortunately, that goal was not achieved. However, it obtained several useful theorems among which the most important one is the main theorem in which it proves that a Lipschitz map F from a compact Riemannian manifold M into a complete Riemannian manifold N admits a smooth approximation Fε via immersions if the map has no singular point in the sense of Clarke by applying the non-smooth analysis established by F. H. Clarke([6,7]). It follows that if a bi-Lipschitz map between two compact manifolds and its inverse map have no singular point in the sense of Clarke, then they are diffeomorphism. As an application of the main theorem, it proves two differentiable sphere theorem by constructing a Lipschitz map between two topological spheres and then proving that the map has no singular point in the sense of Clarke.The main contribution of this paper is to provide a detailed proof of the main theorem in [17]. In [17], the Lipschitz map F from M to N has smooth approximation Fε from M to Rm. By composed with nearest point projection, we have fη from M to N, which is smooth approximation of F. However, [17] doesn’t prove why fη satisfies the conclusion in the main theorem which is completed in this paper. Moreover, this paper corrects some errors and explains some vague explanations of [17] in detail.