Average scalar curvature

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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 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 R d\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 K d\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.