Editing Weyl tensor (section)

==Definition==
{{See also|Ricci decomposition}}
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) valence tensor (by contracting with the metric). The (0,4) valence Weyl tensor is then {{Harv|Petersen|2006|p=92}}

:<math>C = R - \frac{1}{n-2}\left(\mathrm{Ric} - \frac{s}{n}g\right) {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g - \frac{s}{2n(n - 1)}g {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g</math>

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 <math>h {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} k</math> denotes the [[Kulkarni–Nomizu product]] of two symmetric (0,2) tensors:

:<math>\begin{align} 
  (h {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} k)\left(v_1, v_2, v_3, v_4\right) =\quad
          &h\left(v_1, v_3\right)k\left(v_2, v_4\right) + h\left(v_2, v_4\right)k\left(v_1, v_3\right) \\
    {}-{} &h\left(v_1, v_4\right)k\left(v_2, v_3\right) - h\left(v_2, v_3\right)k\left(v_1, v_4\right)
\end{align}</math>

In tensor component notation, this can be written as 
:<math>\begin{align} 
  C_{ik\ell m} = R_{ik\ell m}
      +{} &\frac{1}{n - 2} \left(R_{im}g_{k\ell} - R_{i\ell}g_{km} + R_{k\ell}g_{im} - R_{km}g_{i\ell} \right) \\
    {}+{} &\frac{1}{(n - 1)(n - 2)} R \left(g_{i\ell}g_{km} - g_{im}g_{k\ell} \right).\ 
\end{align}</math>

The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric.

The decomposition ({{EquationNote|1}}) expresses the Riemann tensor as an [[orthogonal]] [[direct sum of vector bundles|direct sum]], in the sense that
:<math>|R|^2 = |C|^2 + \left|\frac{1}{n - 2}\left(\mathrm{Ric} - \frac{s}{n}g\right) {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g\right|^2 + \left|\frac{s}{2n(n - 1)}g {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g\right|^2.</math>

This decomposition, known as the [[Ricci decomposition]], expresses the Riemann curvature tensor into its [[irreducible representation|irreducible]] components under the action of the [[orthogonal group]].{{sfn|Singer|Thorpe|1969}} In dimension 4, the Weyl tensor further decomposes into invariant factors for the action of the [[special orthogonal group]], the self-dual and antiself-dual parts ''C''<sup>+</sup> and ''C''<sup>−</sup>.

The Weyl tensor can also be expressed using the [[Schouten tensor]], which is a trace-adjusted multiple of the Ricci tensor,
:<math>P = \frac{1}{n - 2}\left(\mathrm{Ric} - \frac{s}{2(n-1)}g\right).</math>

Then
:<math>C = R - P {~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~} g.</math>

In indices,<ref>{{Harvnb|Grøn|Hervik|2007|loc=p. 490}}</ref>
:<math>C_{abcd} = R_{abcd} - \frac{2}{n - 2}\left(g_{a[c}R_{d]b} - g_{b[c}R_{d]a}\right) + \frac{2}{(n - 1)(n - 2)}R~g_{a[c}g_{d]b}</math>

where <math>R_{abcd}</math> is the Riemann tensor, <math>R_{ab}</math> is the Ricci tensor, <math>R</math> is the Ricci scalar (the scalar curvature) and brackets around indices refers to the [[Antisymmetric tensor|antisymmetric part]].  Equivalently,
:<math>{C_{ab}}^{cd} = {R_{ab}}^{cd} - 4S_{[a}^{[c}\delta_{b]}^{d]}</math>

where ''S'' denotes the [[Schouten tensor]].