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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 ...
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature.These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations.
The parts S, E, and C of the Ricci decomposition of a given Riemann tensor R are the orthogonal projections of R onto these invariant factors, and correspond (respectively) to the Ricci scalar, the trace-removed Ricci tensor, and the Weyl tensor of the Riemann curvature tensor. In particular,
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
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.
In the same way, the Schur lemma for the Riemann tensor is employed to study convergence of Ricci flow in higher dimensions. This goes back to Gerhard Huisken 's extension of Hamilton's work to higher dimensions, [ 2 ] where the main part of the work is that the Weyl tensor and the semi-traceless Riemann tensor become zero in the long-time limit.
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.