# Definition:Riemann-Christoffel Tensor

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## 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.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Riemann-Christoffel curvature tensor** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Riemann-Christoffel curvature tensor**