Comparison and Rigidity Theorems in Semi-Riemannian Geometry

. The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi--Riemannian manifolds of arbitrary index, using one--sided bounds on the Riemann tensor which in the Riemannian case correspond to one--sided bounds on the sectional curvatures. Starting from 2--dimensional rigidity results and using an inductive technique, a new class of gap--type rigidity theorems is proved for semi--Riemannian manifolds of arbitrary index, generalizing those first given by Gromov and Greene--Wu. As applications we prove rigidity results for semi--Riemannian manifolds with simply connected ends of constant curvature. Contents 1. Introduction 2 1.1. Background 3 1.2. Overview of this paper. 4 2. Preliminaries 5 2.1. Basic Formulas and Results 5 2.2. Warped Products and Model Spaces 8 2.3. Two Dimensional Rigidity Results 11 3. Basic Comparison Results 14 3.1. Comparisons for Parallel Hypersurfaces 14 3.2. Comparisons for Tubes 21 4. Rigidi..