On the isometry group and the geometric structure of compact stationary Lorentzian manifolds

TL;DR

This paper investigates the geometric structure of certain compact Lorentzian manifolds with rich symmetry groups, revealing they are essentially products of flat Lorentzian tori and other well-understood manifolds.

Contribution

It classifies the structure of compact stationary Lorentzian manifolds with infinitely many isometry components, showing they are, up to finite cover, products involving flat Lorentzian tori.

Findings

Such manifolds are products or amalgamated products of flat Lorentzian tori and Riemannian or lightlike manifolds.

The isometry group of these manifolds has infinitely many connected components.

The geometric structure can be described explicitly up to a finite cover.

Abstract

We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (resp., lightlike) manifold.