Okumura Model

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} (R i c - \frac{R}{2 (n - 1)} g) \Leftrightarrow R i c = (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_{i j k l} = W_{i j k l} + g_{i k} P_{j l} - g_{j k} P_{i l} - g_{i l} P_{j k} + g_{j l} P_{i k} .

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

g_{i j} \mapsto Ω^{2} g_{i j} \Rightarrow P_{i j} \mapsto P_{i j} - \nabla_{i} Υ_{j} + Υ_{i} Υ_{j} - \frac{1}{2} Υ_{k} Υ^{k} g_{i j},

Further reading