Given a Riemannian or a pseudo-Riemannian manifold. The Ricci tensor or Ricci curvature, denoted by \(\Rc\) or \(\Ric\), is the trace of the curvature endomorphism on its first and last index.
The components of \(\Rc\) are usually denoted by \(R_{ij}\). Thus, we have
\[ R_{ij}=R_{kij}{}^k. \]
Remarks
- The Ricci curvature is a symmetric tensor field.
- By Bianchi identities follows \[ R_{ij}=R_{ik}{}^k{}_j. \]
Examples