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Riemann curvature tensor

In the [mathematical] field of [differential geometry] , the Riemann curvature tensor , or Riemannâ€“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:

:R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w - \nabla_{[u,v]} w

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
:R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w .

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