In coordinates, and denoting the Ricci tensor by R_ij and the scalar curvature by R, the components of the Cotton tensor are

${\displaystyle C_{ijk}=\nabla _{k}R_{ij}-\nabla _{j}R_{ik}+{\frac {1}{2(n-1)}}\left(\nabla _{j}Rg_{ik}-\nabla _{k}Rg_{ij}\right).}$ 

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

${\displaystyle C_{i}^{j}=\nabla _{k}\left(R_{li}-{\frac {1}{4}}Rg_{li}\right)\epsilon ^{klj},}$ 

sometimes called the Cotton–York tensor.

Conformal rescaling edit

Under conformal rescaling of the metric ${\displaystyle {\tilde {g}}=e^{2\omega }g}$  for some scalar function ${\displaystyle \omega }$ . We see that the Christoffel symbols transform as

${\displaystyle {\widetilde {\Gamma }}_{\beta \gamma }^{\alpha }=\Gamma _{\beta \gamma }^{\alpha }+S_{\beta \gamma }^{\alpha }}$ 

where ${\displaystyle S_{\beta \gamma }^{\alpha }}$  is the tensor

${\displaystyle S_{\beta \gamma }^{\alpha }=\delta _{\gamma }^{\alpha }\partial _{\beta }\omega +\delta _{\beta }^{\alpha }\partial _{\gamma }\omega -g_{\beta \gamma }\partial ^{\alpha }\omega }$ 

The Riemann curvature tensor transforms as

${\displaystyle {{\widetilde {R}}^{\lambda }}{}_{\mu \alpha \beta }={R^{\lambda }}_{\mu \alpha \beta }+\nabla _{\alpha }S_{\beta \mu }^{\lambda }-\nabla _{\beta }S_{\alpha \mu }^{\lambda }+S_{\alpha \rho }^{\lambda }S_{\beta \mu }^{\rho }-S_{\beta \rho }^{\lambda }S_{\alpha \mu }^{\rho }}$ 

In ${\displaystyle n}$ -dimensional manifolds, we obtain the Ricci tensor by contracting the transformed Riemann tensor to see it transform as

${\displaystyle {\widetilde {R}}_{\beta \mu }=R_{\beta \mu }-g_{\beta \mu }\nabla ^{\alpha }\partial _{\alpha }\omega -(n-2)\nabla _{\mu }\partial _{\beta }\omega +(n-2)(\partial _{\mu }\omega \partial _{\beta }\omega -g_{\beta \mu }\partial ^{\lambda }\omega \partial _{\lambda }\omega )}$ 

Similarly the Ricci scalar transforms as

${\displaystyle {\widetilde {R}}=e^{-2\omega }R-2e^{-2\omega }(n-1)\nabla ^{\alpha }\partial _{\alpha }\omega -(n-2)(n-1)e^{-2\omega }\partial ^{\lambda }\omega \partial _{\lambda }\omega }$ 

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

${\displaystyle {\widetilde {C}}_{\alpha \beta \gamma }=C_{\alpha \beta \gamma }+(n-2)\partial _{\lambda }\omega {W_{\beta \gamma \alpha }}^{\lambda }}$ 

or using coordinate independent language as

${\displaystyle {\tilde {C}}=C\;+(n-2)\;\operatorname {grad} \,\omega \;\lrcorner \;W,}$ 

where the gradient is contracted with the Weyl tensor W.

Symmetries edit

The Cotton tensor has the following symmetries:

${\displaystyle C_{ijk}=-C_{ikj}\,}$ 

and therefore

${\displaystyle C_{[ijk]}=0.\,}$ 

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

${\displaystyle \delta W=(3-n)C,\,}$ 

where ${\displaystyle \delta }$  is the positive divergence in the first component of W.

• Aldersley, S. J. (1979). "Comments on certain divergence-free tensor densities in a 3-space". Journal of Mathematical Physics. 20 (9): 1905–1907. Bibcode:1979JMP....20.1905A. doi:10.1063/1.524289.
• Choquet-Bruhat, Yvonne (2009). General Relativity and the Einstein Equations. Oxford, England: Oxford University Press. ISBN 978-0-19-923072-3.
• Cotton, É. (1899). "Sur les variétés à trois dimensions". Annales de la Faculté des Sciences de Toulouse. II. 1 (4): 385–438. Archived from the original on 2007-10-10.
• Eisenhart, Luther P. (1977) [1925]. Riemannian Geometry. Princeton, NJ: Princeton University Press. 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 arXiv:gr-qc/0309008