# Average 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*

## Contents

## Definition

### Given data

A compact connected differential manifold with a Riemannian metric .

### Definition part

The **average scalar curvature** of is defined as the volume-averaged value 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.

## Facts

### For surfaces

For the two-dimensional case, viz surfaces, in the special event that the surface is orientable, we can use the Gauss-Bonnet theorem to determine the average value of scalar curvature over the manifold. That is, we use the fact that where denotes the Euler characteristic, which is a topological invariant (and hence, specifically, a property intrinsic to the differential manifold).

Since , this gives:

where denotes the total volume of the manifold.