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  2. Contracted Bianchi identities - Wikipedia

    en.wikipedia.org/wiki/Contracted_Bianchi_identities

    In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.

  3. Ricci curvature - Wikipedia

    en.wikipedia.org/wiki/Ricci_curvature

    This function on the set of unit tangent vectors is often also called the Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor. The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information.

  4. Curvature invariant (general relativity) - Wikipedia

    en.wikipedia.org/wiki/Curvature_invariant...

    To be able to do this is necessary to include higher-order invariants including derivatives of the Riemann tensor but in the Lorentzian case, it is known that there are spacetimes which cannot be distinguished; e.g., the VSI spacetimes for which all such curvature invariants vanish and thus cannot be distinguished from flat space.

  5. Scalar curvature - Wikipedia

    en.wikipedia.org/wiki/Scalar_curvature

    Given a Riemannian metric g, the scalar curvature Scal is defined as the trace of the Ricci curvature tensor with respect to the metric: [1] = ⁡. The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace.

  6. Curvature invariant - Wikipedia

    en.wikipedia.org/wiki/Curvature_invariant

    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.

  7. Kretschmann scalar - Wikipedia

    en.wikipedia.org/wiki/Kretschmann_scalar

    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.

  8. Ricci decomposition - Wikipedia

    en.wikipedia.org/wiki/Ricci_decomposition

    With this convention, the Ricci tensor is a (0,2)-tensor field defined by R jk =g il R ijkl and the scalar curvature is defined by R=g jk R jk. (Note that this is the less common sign convention for the Ricci tensor; it is more standard to define it by contracting either the first and third or the second and fourth indices, which yields a Ricci ...

  9. Ricci calculus - Wikipedia

    en.wikipedia.org/wiki/Ricci_calculus

    [a] [1] [2] [3] It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), tensor calculus or tensor analysis developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. [4]

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