Variations on Gromov's open-dense orbit theorem

TL;DR

This paper extends Gromov's open-dense orbit theorem by showing that local homogeneity on a dense open subset can imply global local homogeneity, with applications to Lorentz manifolds and isometric actions.

Contribution

It provides new conditions under which local homogeneity on dense open sets implies global local homogeneity, strengthening Gromov's theorem.

Findings

Any smooth closed 3D Lorentz manifold with a topologically transitive isometric action is locally homogeneous.

Strengthened conditions for local homogeneity from dense open subsets to entire manifolds.

Enhanced understanding of geometric structures with local symmetry properties.

Abstract

We investigate several situations where the local homogeneity of a geometric structure on a dense open subset of a manifold implies the local homogeneity everywhere. This results in a strengthening of the conclusions in Gromov's open-dense orbit theorem. In particular, we show that any smooth closed 3-dimensional Lorentz manifold with a topologically transitive isometric action must be locally homogeneous.