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

• Arthur L. Besse, Einstein Manifolds. Springer-Verlag, 2007. See Ch.1 §J "Conformal Changes of Riemannian Metrics."
• Spyros Alexakis, The Decomposition of Global Conformal Invariants. Princeton University Press, 2012. Ch.2, noting in a footnote that the Schouten tensor is a "trace-adjusted Ricci tensor" and may be considered as "essentially the Ricci tensor."
• Wolfgang Kuhnel and Hans-Bert Rademacher, "Conformal diffeomorphisms preserving the Ricci tensor", Proc. Amer. Math. Soc. 123 (1995), no. 9, 2841–2848. Online eprint (pdf).
• T. Bailey, M.G. Eastwood and A.R. Gover, "Thomas's Structure Bundle for Conformal, Projective and Related Structures", Rocky Mountain Journal of Mathematics, vol. 24, Number 4, 1191-1217.

See also

• Weyl–Schouten theorem
• Cotton tensor