Scalar curvature

Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows:

When the scalar curvature is positive at a point, the volume of a small ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is larger than it would be in Euclidean space.

The 2-dimensional Riemann curvature tensor has only one independent component, and it can be expressed in terms of the scalar curvature and metric area form. Namely, in any coordinate system, one has

A space form is by definition a Riemannian manifold with constant sectional curvature. Space forms are locally isometric to one of the following types:

The Riemann tensor of an n-dimensional Euclidean space vanishes identically, so the scalar curvature does as well.

Among those who use index notation for tensors, it is common to use the letter R to represent three different things: