Scalar curvature

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

This article defines a scalar function on a manifold, viz a function from the manifold to real numbers. The scalar function may be intrinsic or defined in terms of some other structure/functions

Definition

Let $(M,g)$ be a Riemannian manifold.

In terms of the Ricci curvature tensor

The scalar curvature associated to $(M,g)$ is defined as the trace of the Ricci curvature tensor of its Levi-Civita connection. By trace, we mean trace, when it is written as a symmetric bilinear form in terms of an orthonormal basis for the Riemannian metric.

In terms of the Ricci curvature

The scalar curvature is a scalar function that associates a curvature at every point $p\in M$ as follows. Consider an orthonormal basis $e_{i}$ for $T_{p}(M)$ . Then, the scalar curvature at $p$ is the sum of the Ricci curvatures for all vectors in the orthonormal basis.

In terms of the sectional curvature

The scalar curvature at a point $p\in M$ is defined as follows. Let $e_{i}$ be an orthonormal basis at $p$ . The scalar curvature is then:

$\sum _{1\leq i<j\leq n}2K(e_{i},e_{j})$

where $K(e_{i},e_{j})$ denotes the sectional curvature of the plane spanned by $e_{i}$ and $e_{j}$ .

In terms of the Riemann curvature tensor

The scalar curvature can be viewed as a double-trace of the Riemann curvature tensor. A more explicit way of viewing it is as follows. Let $e_{i}$ be an orthonormal basis at $p$ . The scalar curvature at $p$ is:

$\sum _{1\leq i\leq n,1\leq j\leq n}R(e_{i},e_{j},e_{j},e_{i})$

Related notions

Related metric properties

Facts

Scalar curvature in terms of Ricci curvature

If the manifold has dimension $n$ , and if the Ricci curvature is constant at a point, the scalar curvature is $n$ times the Ricci curvature at that point.

If the sectional curvature is constant at the point, the scalar curvature is $n(n-1)$ times the sectional curvature.