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...Codazzi tensors impose strict conditions on the [[Riemann curvature tensor|curvature tensor]] of the manifold. Also, the second fundamental form of an immersed * Let <math>(M,g)</math> be a space form with constant curvature <math>\kappa.</math> Given any function <math>f</math> on <math>M,</math> t ...

===Parallel Tensors=== An example for recurrent [[Tensor field|tensor]]s are parallel tensors which are defined by ...

...h>{R^\rho}_\mu</math> is the [[Ricci tensor]], <math>R</math> the [[scalar curvature]], and <math>\nabla_\rho</math> indicates [[covariant differentiation]]. Start with the [[Curvature form#Bianchi identities|Bianchi identity]]<ref name="syngeandschild">{{cite ...

...ed by the [[Riemann curvature tensor]], or in other words, behavior of the curvature tensor under parallel translation determines the metric. ...l|last=Ambrose|first=W.|year=1956|title=Parallel Translation of Riemannian Curvature|journal=The Annals of Mathematics|publisher=JSTOR|volume=64|issue=2|page=33 ...

For ''tensors'', an appropriate generalization is needed. The proper definition for a rep ...s (<math>B^\alpha_\beta</math> is a matrix representation of the surface's curvature shape operator) ...

* the [[sectional curvature]], which assigns to every 2-dimensional linear subspace <math>V</math> of < * the [[Riemann curvature tensor]], which is a multilinear map <math>\operatorname{Rm}_p : T_p M \tim ...

The [[Riemann curvature tensor]] is defined in terms of the [[Levi-Civita connection]] <math>\Gamma Finally, the variation of the [[Ricci curvature tensor]] follows by contracting two indices, proving the identity ...

...<ref>{{cite web|author=[[Jean Gallier]] and Dan Guralnik|title=Chapter 13: Curvature in Riemannian Manifolds|url=http://www.cis.upenn.edu/~cis610/diffgeom5.pdf| [[Category:Tensors in general relativity]] ...

...is that of the [[antisymmetric tensor]] or [[alternating form]]. Symmetric tensors occur widely in [[engineering]], [[physics]] and [[mathematics]]. The space of all symmetric tensors of order ''k'' defined on ''V'' is often denoted by ''S''^''k''(' ...

The foregoing relies on the formalism of [[tensors]], including the [[summation convention]] and the [[Raising and lowering in ...}_{\alpha }</math> be its [[trace (linear algebra)|trace]], that is [[mean curvature]]. Furthermore, let the [[internal energy]] density per unit mass function ...

...=\overline{\Phi}_{20}\,,\Phi_{12}=\overline{\Phi}_{21}\}</math> and the NP curvature scalar <math>\Lambda</math>. Physically, Ricci-NP scalars are related with ...Lambda=\frac{R}{24}</math> with <math>R</math> being the ordinary [[scalar curvature]] of the spacetime metric <math>g_{ab}=-l_a n_b - n_a l_b +m_a \bar{m}_b ...

*[[Riemann curvature tensor]] [[Category:Tensors]] ...

.... They proved in 1975 that any smooth function can be realized as a scalar curvature if it becomes negative somewhere on the manifold. Their further research de ...9939-1972-0290309-X|title=Surfaces of revolution with monotonic increasing curvature and an application to the equation <math>\Delta u = 1 - Ke^{\ u}</math> on ...

...ne f(R) gravity]], whose Lagrangian is an arbitrary function of a [[scalar curvature]] <math>R</math> of <math>\Gamma</math>, is considered. * C. Karahan, A. Altas, D. Demir, Scalars, vectors and tensors from metric-affine gravity, ''General Relativity and Gravitation'' '''45''' ...

The Alena Tensor is a class of [[Stress–energy tensor|energy-momentum tensors]] that allows for equivalent description and analysis of physical systems i ...ecause instead of a field and the forces associated with it, there will be curvature. This would mean, although it is not currently the dominant view in physics ...

...th>g_{\alpha \beta} = e^I_\alpha e^J_\beta \eta_{IJ}.</math> We define the curvature as The [[Ricci scalar]] of this curvature is given by <math>e_I^\alpha e_J^\beta {\Omega_{\alpha \beta}}^{IJ}</math>. ...

...dered as a testing ground to investigate the effects of introducing higher-curvature terms in the context of [[AdS/CFT correspondence]]. | title = Tensors, Differential Forms, and Variational Principles ...

...or spacelike components of [[four-vector]]s and four-dimensional spacetime tensors. In all equations, the [[Einstein notation|summation convention]] is used o ...rential form]]s, specifically as an adjoint bundle-valued [[curvature form|curvature 2-form]] (note that fibers of the adjoint bundle are the '''su'''(3) [[Lie ...

{{redirect|Tensor index notation|a summary of tensors in general|Glossary of tensor theory}} ...ittʃi}})--> constitutes the rules of index notation and manipulation for [[tensors]] and [[tensor fields]] on a [[differentiable manifold]], with or without a ...

The dual [[Riemann curvature tensor]] of the dual graviton is defined as follows:<ref name="Bekaert" /> and the dual [[Ricci curvature]] tensor and [[scalar curvature]] of the dual graviton become, respectively ...