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

$C_{ijk}={\mathrm{\nabla }}_k{R}_{ij}-{\mathrm{\nabla }}_j{R}_{ik}+\frac{1}{2(n-1)}({\mathrm{\nabla }}_{j}R{g}_{ik}-{\mathrm{\nabla }}_{k}R{g}_{ij}).$

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

$C_i^j={\mathrm{\nabla }}_k(R_{li}-\frac{1}{4}R{g}_{li}){ϵ}^{klj},$

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

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

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

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

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

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

Similarly the Ricci scalar transforms as

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

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

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

or using coordinate independent language as

$\tilde{C}=C+\mathrm{grad}\omega ⌟W,$

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