Weyl tensor

From Wikipedia, the free encyclopedia

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the traceless component of the Riemann curvature tensor. In other words, it is 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 dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and h O k denotes the Kulkarni-Nomizu product of two symmetric (0,2) tensors:

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

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 the (1,3) valent Weyl tensor satisfies 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.

The Weyl tensor is given in components by

$C_{abcd}=R_{abcd}-\frac{2}{n-2}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})+\frac{2}{(n-1)(n-2)}R~g_{a[c}g_{d]b}$

where $R a b c d$ is the Riemann tensor, $R a b$ is the Ricci tensor, $R$ is the Ricci scalar (the scalar curvature) and $[]$ refers to the antisymmetric part.