Total scalar curvature

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This article defines a notion of curvature for a differential manifold equipped with a Riemannian metric

This article defines a scalar value (viz, a real number) associated with a Riemannian manifold. This real number depends both on the underlying differential manifold and the Riemannian metric

Definition

Given data

A compact connected differential manifold M with a Riemannian metric g.

Definition part

The total scalar curvature of M is defined as the integral over the volume of the scalar curvature over the manifold. That is, if R denotes the scalar curvature and d\mu the volume element, we have that:

 r = \int R d\mu

Here, the volume element d\mu is the natural choice of volume-element arising from the Riemannian metric.

Note that if we divide by the total volume, we get the average scalar curvature.

Further information: average scalar curvature