# Total scalar curvature

*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 with a Riemannian metric .

### Definition part

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

Here, the volume element 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`