Average scalar curvature

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

The article defines a scalar value (that is, a single real number) associated with a differential manifold. The scalar value may be dependent on certain additional structure with which the differential manifold is equipped

Definition

Given data

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

Definition part

The average scalar curvature of $M$ is defined as the volume-averaged value of the scalar curvature over the manifold. That is, if $R$ denotes the scalar curvature and $d\mu$ the volume element, we have that:

$r={\frac {\int Rd\mu }{\int d\mu }}$

Here, the volume element $d\mu$ 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 $\int Kd\mu =2\pi \chi (M)$ where $\chi$ denotes the Euler characteristic, which is a topological invariant (and hence, specifically, a property intrinsic to the differential manifold).

Since $R=2K$, this gives:

$r={\frac {4\pi \chi (M)}{A}}$

where $A$ denotes the total volume of the manifold.