Scalar curvature

Summary

The scalar curvature of a Riemannian manifold is the function obtained by taking the trace of the Ricci curvature with respect to the metric. It is the simplest curvature invariant: a single number at each point. It measures how the volume of a small geodesic ball differs from the volume of a Euclidean ball of the same radius. Questions such as which manifolds admit metrics of positive scalar curvature are a major theme of geometry and topology.