Pseudo-Riemannian Weakly Symmetric Manifolds

TL;DR

This paper extends the theory of weakly symmetric Riemannian manifolds to pseudo-Riemannian cases, exploring properties like homogeneity, geodesic completeness, and symmetries, with new examples illustrating the similarities and challenges.

Contribution

It demonstrates that many Riemannian results hold for pseudo-Riemannian manifolds, with some requiring extra conditions, and provides new examples highlighting these extensions and limitations.

Findings

Many Riemannian properties extend to pseudo-Riemannian cases

Additional hypotheses are needed for some results

Examples show both parallels and challenges in the extension

Abstract

There is a well developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit property, weak symmetries, and the structure of the nilradical of the isometry group. Also, we give a number of examples of weakly symmetric pseudo-Riemannian manifolds, some mirroring the Riemannian case and some indicating the problems in extending Riemannian results to weakly symmetric pseudo-Riemannian spaces.