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

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

• Flat metric

Weaker properties

• Einstein metric
• Locally homogeneous metric
• Constant-scalar curvature metric

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