Curvature and rigidity in metric spaces

Abstract

In this thesis, we are working the rigidity problems in comparison geometry, in particular "metric geometry" of manifolds with curvature bounds. e.g., manifolds with sectional, Ricci or flag curvature bounded below or above. We are also interested in determining the metric structure of manifolds with curvature bounds. For the sake of convenience we organize the thesis as follows. In chapter 1, we introduce the metric geometry. We also consider the historical remarks on Alexandrov and Finsler manifolds. In chapter 2, on an Alexandrov space with curvature bound, we prove that a curvature takes the extreme value over some specially constructed surfaces if and only if each of the surfaces is totally geodesic and locally isometric to a surface with constant curvature. In chapter 3, we discuss volume on Finsler manifolds and tangent bundles. In chapter 4, we consider the compact perturbation on Finsler manifolds. For a Euclidean space or a Minkowski space, we change the metric in a compact subset and show that the resulting Finsler manifold is isometric to the original standard space under certain conditions. We assume that the mean tangent curvature vanishes and the metric satisfies some curvature conditions or have no conjugate points.