# Locally homogeneous metric

From Diffgeom

*This article defines a property that makes sense for a Riemannian metric over a differential manifold*

## Contents

## Definition

### Given data

A differential manifold equipped with a Riemannian metric .

### Definition part

is said to be **locally homogeneous** if for any we can find neighbourhoods and of those and a Riemannian isometry between and that takes to .

## Relation with other properties

### Stronger properties

- Constant-curvature metric
- Homogeneous metric: The two properties become equivalent when the manifold is simply connected