Schouten tensor

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

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

where Ric is the Ricci 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 $\Upsilon_i := \Omega^{-1} \partial_i \Omega\, .$

Further reading