Constant-curvature metric

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

Definition

Symbol-free definition

A Riemannian metric on a differemtial 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.

Relation with other properties

Weaker properties

• Einstein metric
• Locally homogeneous metric

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