## FANDOM

The Riemann Curvature Tensor is a tensor in Riemannian Geometry that describes the curvature of a manifold. Specifically, it measures the failure (known as the holonomy) of the ability of parallel transport to retain the direction of a vector in a space. In other words, it can be calculated as:

$R_{\mu\nu\rho}^\sigma=\mbox{d}x^\sigma\left[\nabla_\mu,\nabla_\nu\right]\partial_\sigma$

## IdentitiesEdit this section

The definition of the Riemann Curvature Tensor immediately implies that:

$R_{\mu\nu\rho\sigma}=-R_{\nu\mu\rho\sigma}$

And therefore;

$R_{(\mu\nu)\rho\sigma}=0$

Now, below are some other identities:

$R_{\mu\nu(\rho\sigma)} = 0$ See [1] .

First Bianchi Identity [2] .

Second Bianchi Identity [3] .