# 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 be a Riemannian manifold.

### In terms of the Ricci curvature tensor

The **scalar curvature** associated to 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 as follows. Consider an orthonormal basis for . Then, the scalar curvature at 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 is defined as follows. Let be an orthonormal basis at . The scalar curvature is then:

where denotes the sectional curvature of the plane spanned by and .

### 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 be an orthonormal basis at . The scalar curvature at is:

## Related notions

### Related metric properties

## Facts

### Scalar curvature in terms of Ricci curvature

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

If the sectional curvature is constant at the point, the scalar curvature is times the sectional curvature.