Schouten tensor
In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for n ≥ 3 by: P = 1 n − 2 ( R i c − R 2 ( n − 1 ) g ) ⇔ R i c = ( n − 2 ) P + J g , {\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, J = 1 2 ( n − 1 ) R {\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.