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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).
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or ...
The Riemann curvature tensor is defined in terms of the Levi-Civita connection as = +. Its variation is = + +. While the connection is not a tensor, the difference between two connections is, so we can take its covariant derivative
The Riemann tensor is a multilinear operator of fourth rank acting on tangent vectors. However, it can also be considered a linear operator acting on bivectors , and as such it has a characteristic polynomial , whose coefficients and roots ( eigenvalues ) are polynomial scalar invariants.
Although individually, the Weyl tensor and Ricci tensor do not in general determine the full curvature tensor, the Riemann curvature tensor can be decomposed into a Weyl part and a Ricci part. This decomposition is known as the Ricci decomposition, and plays an important role in the conformal geometry of Riemannian manifolds.
where is the Ricci curvature tensor and is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor). The Ricci tensor vanishes in vacuum spacetimes (such as the Schwarzschild solution mentioned above), and hence there the Riemann tensor and the Weyl tensor coincide, as do their invariants.
Mathematically, the tidal force in general relativity is described by the Riemann curvature tensor, [1] and the trajectory of an object solely under the influence of gravity is called a geodesic. The geodesic deviation equation relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics.
Bach tensor, for a sometimes useful tensor generated by via a variational principle. Carminati-McLenaghan invariants, for a set of polynomial invariants of the Riemann tensor of a four-dimensional Lorentzian manifold which is known to be complete under some circumstances.