: Mathématiques: Riemann-Christoffel tensor

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topology > differential topology > differentiable manifold > Riemannian manifold > Riemann-Christoffel tensor

geometry > differential geometry > Riemannian geometry > Riemannian manifold > Riemann-Christoffel tensor

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Riemann-Christoffel tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Riemann_curvature_tensor)

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