# Constant-curvature metric

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

*This is the property of the following curvature being constant:* sectional curvature

## Contents

## Definition

### Symbol-free definition

A Riemannian metric on a differential manifold is termed a **constant-curvature metric** if, for any section of the manifold, the sectional curvature is constant at all points, and moreover, this constant value is the same for all sections.

Equivalently, a Riemannian metric on a differential manifold is termed a constant-curvature metric if it satisfies the following, termed the **axiom of free mobility**, namely: given any two points in the manifold, and any orthonormal bases for the tangent spaces at the two points, there are neighbourhoods of the two points and a Riemannian isometry from one to the other, that maps one orthonormal basis to the other.

## Relation with other properties

### Stronger properties

### Weaker properties

For manifolds of dimension upto three, constant-curvature metrics are the same as Einstein metrics.