Definition:Riemann-Christoffel Tensor
(Redirected from Definition:Riemann Curvature 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): Riemann-Christoffel curvature tensor
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Riemann-Christoffel curvature tensor