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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

Contents

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 \le i < j \le 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 \le i \le n, 1 \le j \le n} R(e_i,e_j,e_j,e_i)

Related notions

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.