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In [[differential geometry]], the term '''curvature tensor''' may refer to: ...e [[Riemann curvature tensor]] of a Riemannian manifold — see also [[Curvature of Riemannian manifolds]]; ...

...ally the [[Riemann tensor]], the [[Weyl tensor]], the [[Ricci tensor]] and tensors formed from these by the operations of taking [[Duality (mathematics)|dual] ==Types of curvature invariants== ...

...the first and third indices of the Riemann tensor), ''R'' is the [[scalar curvature]], ''g'' is the [[Riemannian metric]], <math>J=\frac{1}{2(n-1)}R</math> is The [[Weyl tensor]] equals the [[Riemann curvature tensor]] minus the [[Kulkarni&ndash;Nomizu product]] of the Schouten tensor ...

If ''h'' and ''k'' are symmetric {{nowrap|(0, 2)}}-tensors, then the product is defined via:<ref>Some authors include an overall facto ...urvature]]) and two-dimensional smooth Riemannian manifolds, the [[Riemann curvature tensor]] has a simple expression in terms of the Kulkarni–Nomizu product of ...

...[Weyl tensor|Weyl]] and [[Ricci tensor|Ricci]] tensors — which represent [[curvature]], hence the name — and possibly operations on them such as [[Tensor contra Certain invariants formed from these curvature tensors play an important role in classifying [[spacetime]]s. Invariants are actual ...

...braically independent]] of the [[Weyl tensor]].<ref>P. Szekeres, Conformal Tensors. Proceedings of the Royal Society of London. Series A, Mathematical and Phy ...iven in terms of the [[Ricci tensor]] ''<math>R_{ab}</math>'' and [[scalar curvature]] ''<math>R</math>'' by ...

General four-tensors are usually written in [[tensor index notation]] as ...erest are first order ([[four-vector]]s) or second order, but higher-order tensors occur. Examples are listed next. ...

...a [[connection form|connection]] on a [[principal bundle]]. The [[Riemann curvature tensor]] in [[Riemannian geometry]] can be considered as a special case. Then the '''curvature form''' is the <math>\mathfrak g</math>-valued 2-form on ''P'' defined by ...

{{For|a more elementary discussion|Curvature of space}} ...e different curvatures in different directions, described by the [[Riemann curvature tensor]].]] ...

...s-of-negative-dimensional-tensors.pdf Applications of negative dimensional tensors]," in ''Combinatorial Mathematics and its Applications'', Academic Press (1 === Tensors === ...

...the latter normalisation, the imbedded real projective plane has Gaussian curvature 1. An explicit demonstration of the Riemann and Ricci tensors is given in the ''n''=2 subsection of the article on the [[Fubini-Study met ...

{{Short description|Measure of the curvature of a pseudo-Riemannian manifold}} ...ly how the shape of the body is distorted by the tidal force. The [[Ricci curvature]], or [[trace (linear algebra)|trace]] component of the Riemann tensor cont ...

...icci decomposition''' is a way of breaking up the [[Riemann tensor|Riemann curvature tensor]] of a [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian man ...be a Riemannian or pseudo-Riemannian ''n''-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention ...

...nati–McLenaghan invariants''' or '''CM scalars''' are a set of 16 scalar [[curvature invariant]]s for the [[Riemann tensor]]. This set is usually supplemented *[[Curvature invariant]], for more about curvature invariants in (semi)-Riemannian geometry in general ...

and likewise for tensors with more indices. == Curvature tensors == ...

{{Short description|Mathematical notation for tensors and spinors}} A [[tensor contraction]] (or trace) between two tensors is represented by the repetition of an index label, where one label is cont ...

...be generated from, e.g., the scalar product of vectors, or by contracting tensors. While the components of the contracted quantities may change under Lorentz ...etime from [[general relativity]], which is a contraction of the [[Riemann curvature tensor]] there. ...

...tensor]] vanishes. This follows from the fact that these two second rank tensors stand in a kind of dual relationship; they are the '''trace reverse''' of e ...re tensor]] plus terms built out of the Ricci tensor: the Weyl and Riemann tensors agree, <math>R_{abcd}=C_{abcd}</math>, in some region if and only if it is ...

...s to [[general relativity]], the '''Kretschmann scalar''' is a quadratic [[curvature invariant (general relativity)|scalar invariant]]. It was introduced by [[ \Gamma^{a}{}_{de}\Gamma^{e}{}_{cb}</math> is the [[Riemann curvature tensor]] and <math>\Gamma</math> is the [[Christoffel symbol]]. Because it ...

...he [[Riemann tensor]] of a [[pseudo-Riemannian manifold]] into lower order tensors with properties similar to the [[electric field]] and [[magnetic field]]. *[[Curvature invariant]] ...