Ricci curvature tensor

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This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (0,2)

Description

Given data

A differential manifold M with a linear connection \nabla on it.

Definition part

The Ricci curvature tensor is a (0,2)-tensor that takes as input two vector fields and outputs a scalar function, as follows.

Let X and Y be two vector fields. Then consider the map that sends a vector field Z to R(X,Z)Y (here R denotes the Riemann curvature tensor).

This is a linear map. The Ricci curvature function Ric(X,Y) is defined as the trace of this map.

Explicitly, it is given by:

\operatorname{Tr}(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y)

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.