# A local version of the Myers-Steenrod Theorem

## TL;DR

This paper extends the Myers-Steenrod theorem to local isometry groups on certain Riemannian manifolds, providing new regularity results for locally homogeneous metrics.

## Contribution

It establishes a local version of the Myers-Steenrod theorem for topological groups of isometries on $ ext{C}^{k, extalpha}$-Riemannian manifolds, with applications to metric regularity.

## Key findings

- Proved a local Myers-Steenrod theorem for $ ext{C}^{k, extalpha}$-manifolds.
- Derived regularity results for locally homogeneous Riemannian metrics.
- Extended classical isometry group results to local topological groups.

## Abstract

We prove the Myers-Steenrod theorem for local topological groups of isometries acting on pointed $\mathcal{C}^{k,\alpha}$-Riemannian manifolds, with $k+\alpha>0$. As an application, we infer a new regularity result for a certain class of locally homogeneous Riemannian metrics.