Semisimple Weakly Symmetric Pseudo--Riemannian Manifolds

TL;DR

This paper classifies semisimple weakly symmetric pseudo-Riemannian manifolds, focusing on their structure, isotropy representations, and signatures, including Lorentz and trans-Lorentz cases, extending understanding of these geometric spaces.

Contribution

It provides a comprehensive classification of such manifolds based on the semisimple Lie group structure and their invariant metrics, including specific Lorentzian signatures.

Findings

Classification of weakly symmetric pseudo-Riemannian manifolds with semisimple symmetry groups.

Explicit description of isotropy representations and metric signatures.

Identification of manifolds with Lorentz and trans-Lorentz signatures.

Abstract

We develop the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derive the classification from the cases where $G$ is compact, and then we discuss the (isotropy) representation of $H$ on the tangent space of $G/H$ and the signature of the invariant pseudo--riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans--Lorentz (conformal Lorentz) signature $(n-2,2)$.