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
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:
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.