Conformal rescaling

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

The Cotton tensor has the following symmetries:

${\displaystyle C_{ijk}=-C_{ikj}}$

and therefore^{[clarification needed]}

${\displaystyle C_{[ijk]}=0.}$

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.