# Definition:Riemann-Christoffel Tensor

## Definition

A Riemann-Christoffel tensor is a tensor field which expresses the curvature of a Riemannian manifold.

The Riemann-Christoffel tensor is given in terms of the Levi-Civita connection $\nabla$ by:

$R \left({u, v}\right) w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{\left[{u, v}\right]} w$

where $\left[{u, v}\right]$ is the Lie bracket of vector fields.

It measures the extent to which the metric tensor is not locally isometric to that of Euclidean space.

## Also known as

The Riemann-Christoffel tensor is also known as the Riemann curvature tensor or the Riemann-Christoffel curvature tensor.

## Source of Name

This entry was named for Bernhard Riemann and Elwin Bruno Christoffel.

## Historical Note

The concept of a Riemann-Christoffel tensor was originated by Bernhard Riemann in his application of a Riemannian manifold to the question of heat conduction.