Riemann Curvature Tensor

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:

$R_{\mu \nu \rho }^{\sigma }={\mbox{d}}x^{\sigma }\left[\nabla _{\mu },\nabla _{\nu }\right]\partial _{\sigma }$

Identities

Correction to Proof of the Zeroth Bianchi Identity

More on Bianchi Identities

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 .

First Bianchi Identity .

Second Bianchi Identity .