Category:Definitions/Riemann-Christoffel Tensors

This category contains definitions related to Riemann-Christoffel Tensors.
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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:

$\map R {u, v} w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{\sqbrk {u, v} } w$

where $\sqbrk {u, v}$ 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.