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Weyl tensor
 
 
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In differential geometry, the Weyl curvature tensor, named after Hermann Weyl
Hermann Weyl

Hermann Klaus Hugo Weyl was a Germany mathematician. Although much of his working life was spent in Z?rich, Switzerland and then Princeton, New Jersey, he is associated with the University of G?ttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski....
, is the traceless component of the Riemann curvature tensor
Curvature tensor

The term curvature tensor is ambiguous in its generality. It could refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;...
. In other words, it is a tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
 that has the same symmetries as the Riemann curvature tensor with the extra condition that metric contraction
Tensor contraction

In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the Bilinear form#different spaces of a finite-dimensional vector space and its dual vector space....
 yields zero.

In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero.

If the Weyl tensor vanishes, then there exists a coordinate system in which the metric tensor is proportional to a constant tensor.






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In differential geometry, the Weyl curvature tensor, named after Hermann Weyl
Hermann Weyl

Hermann Klaus Hugo Weyl was a Germany mathematician. Although much of his working life was spent in Z?rich, Switzerland and then Princeton, New Jersey, he is associated with the University of G?ttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski....
, is the traceless component of the Riemann curvature tensor
Curvature tensor

The term curvature tensor is ambiguous in its generality. It could refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;...
. In other words, it is a tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
 that has the same symmetries as the Riemann curvature tensor with the extra condition that metric contraction
Tensor contraction

In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the Bilinear form#different spaces of a finite-dimensional vector space and its dual vector space....
 yields zero.

In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero.

If the Weyl tensor vanishes, then there exists a coordinate system in which the metric tensor is proportional to a constant tensor. This fact was a key component of Nordström's theory of gravitation
Nordström's theory of gravitation

In theoretical physics, Nordstr?m's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordstr?m, in 1912 and 1913 respectively....
, which was an earlier precursor of general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
.

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
Scalar curvature

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point....
, and h O k denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors:

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
Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane....
 changes to the metric
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
. That is, if g′ = f g for some positive scalar function f 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
Conformally flat

A Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.More formally, let be a pseudo-Riemannian manifold....
 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
Cotton tensor

In differential geometry, the Cotton tensor on a -Riemannian manifold of dimension n is a third-order tensor field concomitant of the metric tensor, like the Weyl curvature....
 is a necessary and sufficient condition for the Riemannian manifold being conformally flat.

The Weyl tensor is given in components by where is the Riemann tensor, is the Ricci tensor, is the Ricci scalar (the scalar curvature) and refers to the antisymmetric part
Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:...
.

See also

Curvature of Riemannian manifolds
Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point....
 Christoffel symbols
Christoffel symbols

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor....
 provides a coordinate expression for the Weyl tensor. Petrov classification
Petrov classification

In differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetry of the Weyl tensor at each Spacetime#Basic concepts in a Lorentzian manifold....
 Weyl curvature hypothesis
Weyl curvature hypothesis

The Weyl curvature hypothesis, which arises in the application of Albert Einstein's general theory of relativity to physical cosmology, was introduced by the British mathematician and theoretical physicist Sir Roger Penrose in an article in 1979 in an attempt to provide explanations for two of the most fundamental issues in physics....
 Weyl scalar Cotton tensor
Cotton tensor

In differential geometry, the Cotton tensor on a -Riemannian manifold of dimension n is a third-order tensor field concomitant of the metric tensor, like the Weyl curvature....