Scalar curvature

Given a pseudo-Riemannian manifold $(M,g)$ with Ricci curvature tensor with components $R_{ab}$, the scalar curvature is the number

Interpretation

From this paper.
It turns out the scalar curvature has a meaning very similar to the Gaussian curvature. If we imagine instead of taking a circle, taking a generalized $D-1$ sphere, i.e. the set of all points a geodesic distance $ϵ$ from a given starting point $x_0^{\mu }$. We can calculate the area of this sphere in flat space, but in curved space the area will deviate from the one we calculated by an amount proportional to the curvature. Thus, we find that the scalar curvature is