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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 mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ...
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...
In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...
In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...
In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...
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
 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: In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...
In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ...
The Kulkarni-Nomizu product is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor. ...
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. In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ...
In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...
In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...
A metric manifold is conformally flat if it can be mapped to flat space by a conformal transformation. ...
In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...
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 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}](http://upload.wikimedia.org/math/2/c/3/2c3676e8cd0f0a0333cf8f872b7711fd.png) where Rabcd is the Riemann tensor, Rab is the Ricci tensor, R is the Ricci scalar (the scalar curvature) and [] refers to the antisymmetric part. In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ...
See also
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