The Schouten tensor is a symmetric (0,2)‑tensor field on a smooth pseudo‑Riemannian manifold $(M,g)$ of dimension $n\ge 2$ that plays a central role in conformal geometry and the decomposition of the Riemann curvature tensor.
Definition
Let $\operatorname{Ric}$ denote the Ricci curvature tensor, $R$ the scalar curvature, and $g$ the metric tensor. The Schouten tensor $P$ is defined by
$$ P ;=; \frac{1}{n-2}\Bigl(\operatorname{Ric} - \frac{R}{2(n-1)},g\Bigr), \qquad n\ge 3, $$
and, for the exceptional case $n=2$, it is defined by
$$ P ;=; \frac{1}{2},K,g, $$
where $K$ is the Gaussian curvature (the scalar curvature divided by 2).
Geometric significance
- Curvature decomposition – On manifolds of dimension $n\ge 3$ the Riemann curvature tensor $R_{ijkl}$ can be uniquely expressed as
$$ R_{ijkl}=W_{ijkl}+g_{ik}P_{jl}-g_{il}P_{jk}-g_{jk}P_{il}+g_{jl}P_{ik}, $$
where $W$ is the Weyl conformal curvature tensor. The tensor $P$ thus encapsulates the trace part of the curvature that is not conformally invariant.
- Conformal transformation law – Under a conformal change of metric $\tilde g=e^{2\omega}g$ with smooth function $\omega$, the Schouten tensor transforms according to
$$ \tilde P = P - abla^2\omega + d\omega\otimes d\omega - \frac{1}{2}|d\omega|^2 g, $$
where $ abla^2\omega$ is the Hessian and $|d\omega|^2$ the squared norm with respect to $g$. This transformation law makes $P$ a convenient intermediate object in the study of conformally invariant differential operators.
- Cotton tensor – The Cotton tensor, which measures the failure of a manifold to be conformally flat in dimension three, is defined as the divergence of the Schouten tensor:
$$ C_{ijk}= abla_i P_{jk} - abla_j P_{ik}. $$
In dimension three the vanishing of the Cotton tensor is equivalent to conformal flatness.
Historical note
The tensor is named after the Dutch mathematician Jan Arnoldus Schouten (1883–1971), who contributed significantly to tensor analysis and differential geometry.
Applications
- Conformal geometry and the study of conformally invariant equations (e.g., the Yamabe problem).
- Construction of conformal invariants such as the $Q$-curvature.
- Analysis of geometric flows that preserve or modify conformal structures.
References
- J. A. Schouten, Ricci‑Calculus, Springer, 1954.
- M. G. Eastwood, “The Schouten tensor and conformal geometry,” Proceedings of the 1999 AMS–IMS–SIAM Joint Summer Research Conference, 2000.
- S. Y. Cheng and S. T. Yau, “Differential equations on Riemannian manifolds and their geometric applications,” Comm. Pure Appl. Math., 1977.