Ricci curvature

2-tensor obtained as a contraction of the Rieman curvature 4-tensor on a Riemannian manifold

The first subsection here is meant as an indication of the definition of the Ricci tensor for readers who are comfortable with linear algebra and multivariable calculus. The later subsections use more sophisticated terminology.

With the use of some sophisticated terminology, the definition of Ricci curvature can be summarized as saying:

The remarkable and unexpected property of Ricci curvature can be summarized as:

This is discussed from the perspective of differentiable manifolds in the following subsection, although the underlying content is virtually identical to that of this subsection.

It is common to abbreviate the above formal presentation in the following style:

The final line includes the demonstration that the bilinear map Ric is well-defined, which is much easier to write out with the informal notation.

It is less immediately obvious that the two terms on the right hand side are orthogonal to each other:

An identity which is intimately connected with this (but which could be proved directly) is that

In this more general situation, the Ricci tensor is symmetric if and only if there exist locally a parallel volume form for the connection.