Comparison and Rigidity Theorems in Semi-Riemannian Geometry

1997/07/25 by Andersson, L., Howard, R.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.dg-ga/9707020

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.

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