Eilenberg–MacLane space

In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric, like the Weyl tensor. The vanishing of the Cotton tensor for n=3 is necessary and sufficient condition for the manifold to be conformally flat, as with the Weyl tensor for n≥4. For n<3 the Cotton tensor is identically zero. The concept is named after Émile Cotton.

The proof of the classical result that for n = 3 the vanishing of the Cotton tensor is equivalent the metric being conformally flat is given by Eisenhart using a standard 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 Template:Harv.

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 equations and plays a important role in the Hamiltonian formalism of general relativity.

Definition

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

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

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

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

sometimes called the Cotton–York tensor.

Properties

Conformal rescaling

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

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

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

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

The Riemann curvature tensor transforms as

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

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

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

Similarly the Ricci scalar transforms as

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

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

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

or using coordinate independent language as

${\tilde {C}}=C\;+\;\operatorname {grad} \,\omega \;\lrcorner \;W,$

where the gradient is plugged into the symmetric part of the Weyl tensor W.

Symmetries

The Cotton tensor has the following symmetries:

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

and therefore

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

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

$\delta W=(3-n)C,\,$

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

References

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• 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