Comparison and rigidity theorems in semi-Riemannian geometry (Q1281364)

The comparison theory for the Riccati equation has proved to be a very powerful tool in the comparison geometry for Riemannian manifolds. In the paper under review, the authors generalize the theory to semi-Riemannian manifolds of arbitrary index with one-sided bound on the Riemann curvature tensor. The comparison theory developed in the paper is first used to give a new class of gap-type rigidity theorems which, in some cases, extend those of Gromov and Greene-Wu. The theory is then used to prove rigidity theorems for semi-Riemannian manifolds with simply connected ends of constant curvature. The self-contained paper is very well-written and is an indispensable reference for anyone working in the field of comparison geometry.