Editing Cotton tensor

{{No footnotes|date=July 2017}}

In [[differential geometry]], the '''Cotton tensor''' on a (pseudo)-[[Riemannian manifold]] of dimension ''n'' is a third-order [[tensor field|tensor]] concomitant of the [[metric tensor|metric]]. The vanishing of the Cotton tensor for {{nowrap|1=''n'' = 3}} is [[necessary condition|necessary]] and [[sufficient condition]] for the manifold to be locally [[conformally flat]]. By contrast, in dimensions {{nowrap|''n'' ≥ 4}},
the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient   condition in these higher dimensions is the vanishing of the [[Weyl tensor]], while the Cotton tensor just becomes a constant times 
the divergence of the Weyl tensor. For {{nowrap|''n'' < 3}} the Cotton tensor is identically zero. The concept is named after [[Émile Cotton]].

The proof of the classical result that for {{nowrap|1=''n'' = 3}} the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by [[Luther P. Eisenhart|Eisenhart]] using a standard [[integrability condition|integrability]] argument. This tensor density is  uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics, as shown by {{Harv|Aldersley|1979}}.

Recently, the study of three-dimensional spaces is becoming of great interest, because the Cotton tensor restricts the relation between the Ricci tensor and the [[energy–momentum tensor]] of matter in the [[Einstein equation]]s and plays an important role in the [[Hamiltonian formalism]] of [[general relativity]].

== Definition ==

In coordinates, and denoting the [[Ricci tensor]] by ''R''<sub>''ij''</sub> and the scalar curvature by ''R'', the components of the Cotton tensor are

:<math>C_{ijk} = \nabla_{k} R_{ij} - \nabla_{j} R_{ik} + \frac{1}{2(n-1)}\left( \nabla_{j}Rg_{ik} -  \nabla_{k}Rg_{ij}\right).</math>

The Cotton tensor can be regarded as a vector valued [[Differential form|2-form]], and for ''n''&nbsp;=&nbsp;3 one can use the [[Hodge star operator]] to convert this into a second order trace free tensor density

:<math>C_i^j = \nabla_{k} \left( R_{li} - \frac{1}{4} Rg_{li}\right)\epsilon^{klj},</math>

sometimes called the ''Cotton–[[James W. York|York]] tensor''.

==Properties==

===Conformal rescaling===
Under conformal rescaling of the metric <math>\tilde{g} = e^{2\omega} g</math> for some scalar function <math>\omega</math>. We see that the [[Christoffel symbol]]s transform as

:<math>\widetilde{\Gamma}^{\alpha}_{\beta\gamma}=\Gamma^{\alpha}_{\beta\gamma}+S^{\alpha}_{\beta\gamma}</math>

where <math>S^{\alpha}_{\beta\gamma}</math> is the tensor

:<math>S^{\alpha}_{\beta\gamma} = \delta^{\alpha}_{\gamma} \partial_{\beta} \omega + \delta^{\alpha}_{\beta} \partial_{\gamma} \omega - g_{\beta\gamma} \partial^{\alpha} \omega</math>

The [[Riemann curvature tensor]] transforms as

:<math>{\widetilde{R}^{\lambda}}{}_{\mu\alpha\beta}={R^{\lambda}}_{\mu\alpha\beta}+\nabla_{\alpha}S^{\lambda}_{\beta\mu}-\nabla_{\beta}S^{\lambda}_{\alpha\mu}+S^{\lambda}_{\alpha\rho}S^{\rho}_{\beta\mu}-S^{\lambda}_{\beta\rho}S^{\rho}_{\alpha\mu}</math>

In <math>n</math>-dimensional manifolds, we obtain the [[Ricci tensor]] by contracting the transformed Riemann tensor to see it transform as

:<math>\widetilde{R}_{\beta\mu}=R_{\beta\mu}-g_{\beta\mu}\nabla^{\alpha}\partial_{\alpha}\omega-(n-2)\nabla_{\mu}\partial_{\beta}\omega+(n-2)(\partial_{\mu}\omega\partial_{\beta}\omega-g_{\beta\mu}\partial^{\lambda}\omega\partial_{\lambda}\omega)</math>

Similarly the [[Ricci scalar]] transforms as

:<math>\widetilde{R}=e^{-2\omega}R-2e^{-2\omega}(n-1)\nabla^{\alpha}\partial_{\alpha}\omega-(n-2)(n-1)e^{-2\omega}\partial^{\lambda}\omega\partial_{\lambda}\omega</math>

Combining all these facts together permits us to conclude the Cotton-York tensor transforms as

:<math>\widetilde{C}_{\alpha\beta\gamma}=C_{\alpha\beta\gamma}+(n-2)\partial_{\lambda}\omega {W_{\beta\gamma\alpha}}^{\lambda}</math>

or using coordinate independent language as

:<math> \tilde{C} = C \; + (n-2) \; \operatorname{grad} \, \omega \; \lrcorner \; W,</math>

where the gradient is contracted with the [[Weyl tensor]]&nbsp;''W''.

===Symmetries===
The Cotton tensor has the following symmetries:

:<math>C_{ijk} = - C_{ikj} \, </math>

and therefore

:<math>C_{[ijk]} = 0. \, </math>

In addition the Bianchi formula for the [[Weyl tensor]] can be rewritten as

:<math>\delta W = (3-n) C, \, </math>

where <math>\delta</math> is the positive divergence in the first component of ''W''.

==References==
{{reflist}}
*{{Cite journal |first=S. J. |last=Aldersley |title=Comments on certain divergence-free tensor densities in a 3-space |journal=[[Journal of Mathematical Physics]] |volume=20 |issue=9 |pages=1905–1907 |year=1979 |doi=10.1063/1.524289 |bibcode = 1979JMP....20.1905A |doi-access=free }}
*{{Cite book |first=Yvonne |last=Choquet-Bruhat |authorlink=Yvonne Choquet-Bruhat|title=General Relativity and the Einstein Equations |publisher=[[Oxford University Press]] |location=Oxford, England |year=2009 |isbn=978-0-19-923072-3 }}
*{{Cite journal |first=É. |last=Cotton |authorlink=Émile Cotton |title=Sur les variétés à trois dimensions |journal=[[Annales de la Faculté des Sciences de Toulouse]] |series=II |volume=1 |issue=4 |pages=385–438 |year=1899 |url=http://www.numdam.org/numdam-bin/fitem?id=AFST_1899_2_1_4_385_0 |url-status=dead |archiveurl=https://web.archive.org/web/20071010003140/http://www.numdam.org/numdam-bin/fitem?id=AFST_1899_2_1_4_385_0 |archivedate=2007-10-10 }}
*{{Cite book |first=Luther P. |last=Eisenhart|authorlink=Luther Eisenhart |title=Riemannian Geometry |publisher=[[Princeton University Press]] |location=Princeton, NJ |orig-date=1925 |year=1977 |isbn=0-691-08026-7 }}
* A. Garcia, F.W. Hehl, C. Heinicke, A. Macias (2004) "The Cotton tensor in Riemannian spacetimes", [[Classical and Quantum Gravity]] 21: 1099–1118, Eprint [https://arxiv.org/abs/gr-qc/0309008 arXiv:gr-qc/0309008]

[[Category:Riemannian geometry]]
[[Category:Tensors in general relativity]]
[[Category:Tensors]]