![]() ![]() 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 can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. locally isometric to the Euclidean space. A Riemannian manifold has zero curvature if and only if it is flat, i.e. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). 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.
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