Comparison and Rigidity Theorems in Semi-Riemannian Geometry

The comparison theory for the Riccati equation satis- ed 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 cor- respond to one{sided bounds on the sectional curvatures. Starting from 2{dimensional rigidity results and using an inductive tech- nique, a new class of gap{type rigidity theorems is proved for semi{ Riemannian manifolds of arbitrary index, generalizing those rst given by Gromov and Greene{Wu. As applications we prove rigid- ity results for semi{Riemannian manifolds with simply connected ends of constant curvature.