Average scalar curvature
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.
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.