Ricci curvature tensor
This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (0,2)
The Ricci curvature tensor is a -tensor that takes as input two vector fields and outputs a scalar function, as follows.
This is a linear map. The Ricci curvature function is defined as the trace of this map.
Explicitly, it is given by:
The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.
For a Riemannian or pseudo-Riemannian manifold
Further information: Ricci curvature tensor of Levi-Civita connection
For a Riemannian manifold or pseudo-Riemannian manifold, we can give the Levi-Civita connection, a natural choice of linear connection. The Ricci curvature tensor of this connection, is termed the Ricci curvature tensor of the Riemannian or pseudo-Riemannian manifold.