Comparison and Rigidity Theorems in Semi-Riemannian Geometry

TL;DR

This paper generalizes comparison and rigidity theorems from Riemannian to semi-Riemannian geometry, establishing new gap-type rigidity results using Riccati equation bounds and inductive techniques.

Contribution

It extends comparison theory and rigidity theorems to semi-Riemannian manifolds of arbitrary index, generalizing classical results by Gromov and Greene-Wu.

Findings

Established new gap-type rigidity theorems for semi-Riemannian manifolds.

Proved rigidity results for manifolds with simply connected ends of constant curvature.

Generalized comparison theory using bounds on the Riemann tensor.

Abstract

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.