Total scalar curvature
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