Schouten tensor
From HandWiki
In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for n ≥ 3 by:
- [math]\displaystyle{ P=\frac{1}{n - 2} \left(\mathrm{Ric} -\frac{ R}{2 (n-1)} g\right)\, \Leftrightarrow \mathrm{Ric}=(n-2) P + J g \, , }[/math]
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, [math]\displaystyle{ J=\frac{1}{2(n-1)}R }[/math] 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
- [math]\displaystyle{ R_{ijkl}=W_{ijkl}+g_{ik} P_{jl}-g_{jk} P_{il}-g_{il} P_{jk}+g_{jl} P_{ik}\, . }[/math]
The Schouten tensor often appears in conformal geometry because of its relatively simple conformal transformation law
- [math]\displaystyle{ 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}\, , }[/math]
where [math]\displaystyle{ \Upsilon_i := \Omega^{-1} \partial_i \Omega\, . }[/math]
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
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