Riemann curvature tensor

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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 standard way to express curvature of Riemannian manifolds. It associates a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to a Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.

The curvature tensor is given in terms of the Levi-Civita connection \nabla by the following formula:

where [u,v] is the Lie bracket of vector fields. For each pair of tangent vectors u, v, R(u,v) is a linear transformation of the tangent space of the manifold. It is linear in u and v, and so defines a tensor. Occasionally, the curvature tensor is defined with the opposite sign. If u=\partial/\partial x^i and v=\partial/\partial x^j are coordinate vector fields then [u,v] = 0 and therefore the formula simplifies to

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). The linear transformation w\mapsto R(u,v)w is also called the curvature transformation or endomorphism.

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Geometrical meaning

When a vector in a Euclidean space is parallel transported around a loop, it will always return to its original position. The Riemann curvature tensor directly measures the failure of this in a general Riemannian manifold. This failure is known as the holonomy of the manifold.

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