Schouten tensor

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

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

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, ${\displaystyle J={\frac {1}{2(n-1)}}R}$ 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

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

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

${\displaystyle g_{ij}\mapsto \Omega ^{2}g_{ij}\Rightarrow P_{ij}\mapsto P_{ij}-\nabla _{i}\Upsilon _{j}+\Upsilon _{i}\Upsilon _{j}-{\frac {1}{2}}\Upsilon _{k}\Upsilon ^{k}g_{ij}\,,}$

where ${\displaystyle \Upsilon _{i}:=\Omega ^{-1}\partial _{i}\Omega \,.}$

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