Quasihomogeneous three-dimensional real analytic Lorentz metrics do not exist

TL;DR

This paper proves that real analytic Lorentz metrics in three dimensions cannot be quasihomogeneous unless they are globally homogeneous, providing a classification of possible Lie algebra actions and normal forms in certain cases.

Contribution

It establishes the non-existence of quasihomogeneous 3D real analytic Lorentz metrics and classifies Lie algebra actions for such metrics.

Findings

Quasihomogeneous Lorentz metrics in 3D do not exist.

Complete classification of Lie algebra actions in the semisimple isotropy case.

Normal forms for metrics with semisimple isotropy at the origin.

Abstract

We show that a germ of a real analytic Lorentz metric on ${\bf R}^3$ which is locally homogeneous on an open set containing the origin in its closure is necessarily locally homogeneous. We classifiy Lie algebras that can act quasihomogeneously---meaning they act transitively on an open set admitting the origin in its closure, but not at the origin---and isometrically for such a metric. In the case that the isotropy at the origin of a quasihomogeneous action is semisimple, we provide a complete set of normal forms of the metric and the action.