Tag Info

$$R_{\mu\nu\rho}^\sigma=\mathrm{d}x^\sigma[\nabla_\mu,\nabla_\nu]\partial_\sigma$$
A partial trace of this tensor is a symmetric tensor, namely, the Ricci Curvature Tensor $R_{\mu\nu}=g^{\rho\sigma}R_{\mu\nu\rho\sigma}$, which is very useful in General Relativity, for example. In 4-dimensions, the Riemann Curvature Tensor can completely be described by the Ricci Curvature Tensor and the Weyl Tensor $C_{\mu\nu\rho\sigma}$.