Editing Weyl tensor (section)

===Conformal rescaling===
The Weyl tensor has the special property that it is invariant under [[conformal map|conformal]] changes to the [[metric tensor|metric]]. That is, if <math>g_{\mu\nu}\mapsto g'_{\mu\nu} = f g_{\mu\nu}</math> for some positive scalar function <math>f</math> then the (1,3) valent Weyl tensor satisfies <math>{C'}^{a}_{\ \ bcd} = C^{a}_{\ \ bcd}</math>. For this reason the Weyl tensor is also called the '''conformal tensor'''. It follows that a [[necessary condition]] for a [[Riemannian manifold]] to be [[conformally flat]] is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is [[sufficient condition|sufficient]] as well. In dimension 3 the vanishing of the [[Cotton tensor]] is a necessary and sufficient condition for the Riemannian manifold being conformally flat.  Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a consequence of the existence of [[isothermal coordinates]].

Indeed, the existence of a conformally flat scale amounts to solving the overdetermined partial differential equation
:<math>Ddf - df\otimes df + \left(|df|^2 + \frac{\Delta f}{n - 2}\right)g = \operatorname{Ric}.</math>

In dimension ≥ 4, the vanishing of the Weyl tensor is the only [[integrability condition]] for this equation; in dimension 3, it is the [[Cotton tensor]] instead.