Ricci curvature tensor of Levi-Civita connection

Definition

Let $M$ be a differential manifold and $g$ be a Riemannian metric or pseudo-Riemannian metric on $M$. Let $\mathrm{\nabla }$ be the Levi-Civita connection associated with $g$. The Ricci curvature tensor of $M$ is defined as the Ricci curvature tensor of the Levi-Civita connection.

Explicitly, if $R$ is the Riemann curvature tensor of the Levi-Civita connection:

$\mathrm{R}\mathrm{i}\mathrm{c}(X,Y)=-Tr(Z\mapsto R(X,Z)Y)$

In the particular case of a Riemannian metric, we can choose an orthonormal basis $e_i$ on each tangent space. For a particular tangent space, if the orthonormal basis is $e_i$, the Ricci curvature tensor evaluated at a pair of vectors $X,Y$ is:

$\mathrm{R}\mathrm{i}\mathrm{c}(X,Y)=\sum _{i=1}^{n}g(R(X,e_i)Y,e_i)$

Or is the language of $R$ as a $(0,4)$-tensor:

$\mathrm{R}\mathrm{i}\mathrm{c}(X,Y)=\sum _{i=1}^{n}R(X,e_i,Y,e_i)$

Related notions

• Ricci curvature in a direction is the Ricci curvature tensor $Ric(X,X)$ where $X$ is a unit vector in that direction

Facts

Symmetry

We have:

$Ric(X,Y)=Ric(Y,X)$

This follows from the fact that the Riemann curvature tensor is symmetric in the pair of the first two and last two variables. For full proof, refer: Symmetry of Riemann curvature tensor in variable pairs