FANDOM


Stub

This article is a stub. You can help Mathematics and Physics Wiki by expanding it.

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

Correction to Proof of the Zeroth Bianchi Identity

Correction to Proof of the Zeroth Bianchi Identity

More on Bianchi Identities

More on Bianchi Identities

More on Bianchi Identities-0

More on Bianchi Identities-0

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] .


ReferencesEdit this section