Locally homogeneous metric

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

Definition

Given data

A differential manifold $M$ equipped with a Riemannian metric $g$.

Definition part

$g$ is said to be locally homogeneous if for any $x,y \in M$ we can find neighbourhoods $U_x$ and $U_y$ of those and a Riemannian isometry between $U_x$ and $U_y$ that takes $x$ to $y$.

Relation with other properties

Stronger properties

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

Weaker properties

• Complete metric