Editing Riemann curvature tensor (section)

{{Short description|Tensor field in Riemannian geometry}}
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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 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 that measures the failure of the second [[Covariant derivative|covariant derivatives]] to commute. A Riemannian manifold has zero curvature if and only if it is ''flat'', i.e. locally [[Isometry|isometric]] to the [[Euclidean space]].{{sfn|Lee|2018|p=193}} 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]]. 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 field|Jacobi equation]].