weyl curvature

In differential geometry, the Weyl curvature tensor is the traceless component of the Riemann curvature tensor. In other words, it a tensor that has the same symmetries as the Riemann curvature tensor with the extra condition that its Ricci curvature must vanish. In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a (0,4) valent tensor (by contracting with the metric). The (0,4) valent Weyl tensor is then

W = R - \frac{1}{n-2}\left(Ric - \frac{s}{n}g\right)\circ g - \frac{s}{2n(n-1)}g\circ g

where n is the dimensional of the manifold, g is the metric, Ric is the Ricci tensor, s is the scalar curvature, and hOk denotes the Kulkarni-Nomizu product of two symmetric (0,2) tensors:

math>(h\circ k)(v_1,v_2,v_3,v_4) =	$h(v_1,v_3)k(v_2,v_4)+h(v_2,v_4)k(v_1,v_3)\,$
\| ${}-h(v_1,v_4)k(v_2,v_3)-h(v_2,v_3)k(v_1,v_4)\,$

The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric. The Weyl tensor has the special property that it is invariant under conformal changes to the metric. That is, if g′ = f g for some positive scalar function then W′ = W. 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. It turns out that in dimensions ≥ 4 this condition is 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.

Weyl Curvature

See also