Schouten tensor

In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for n ≥ 3 by:


 * $$P=\frac{1}{n - 2} \left(\mathrm{Ric} -\frac{ R}{2 (n-1)} g\right)\, \Leftrightarrow \mathrm{Ric}=(n-2) P + J g \, ,$$

where Ric is the Ricci tensor (defined by contracting the first and third indices of the Riemann tensor), R is the scalar curvature, g is the Riemannian metric, $$J=\frac{1}{2(n-1)}R$$ is the trace of P and n is the dimension of the manifold.

The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni–Nomizu product of the Schouten tensor with the metric. In an index notation


 * $$R_{ijkl}=W_{ijkl}+g_{ik} P_{jl}-g_{jk} P_{il}-g_{il} P_{jk}+g_{jl} P_{ik}\, . $$

The Schouten tensor often appears in conformal geometry because of its relatively simple conformal transformation law


 * $$g_{ij}\mapsto \Omega^2 g_{ij} \Rightarrow P_{ij}\mapsto P_{ij}-\nabla_i \Upsilon_j + \Upsilon_i \Upsilon_j -\frac12 \Upsilon_k \Upsilon^k g_{ij}\,, $$

where $$ \Upsilon_i := \Omega^{-1} \partial_i \Omega\,. $$