Schouten tensor

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

$P=\frac{1}{n-2}(\mathrm{R}\mathrm{i}\mathrm{c}-\frac{R}{2(n-1)}g)\iff \mathrm{R}\mathrm{i}\mathrm{c}=(n-2)P+Jg,$

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

$g_{ij}\mapsto {\mathrm{\Omega }}^2{g}_{ij}\Rightarrow P_{ij}\mapsto P_{ij}-{\mathrm{\nabla }}_i{\mathrm{{\rm Y}}}_j+{\mathrm{{\rm Y}}}_i{\mathrm{{\rm Y}}}_j-\frac{1}{2}{\mathrm{{\rm Y}}}_k{\mathrm{{\rm Y}}}^k{g}_{ij},$

where ${\mathrm{{\rm Y}}}_i:={\mathrm{\Omega }}^{-1}{\partial }_i\mathrm{\Omega }.$

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