Girish Mahajan (Editor)

Schouten tensor

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

P = 1 n 2 ( R i c R 2 ( n 1 ) g ) 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 = 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 Ω 2 g i j P i j P i j i Υ j + Υ i Υ j 1 2 Υ k Υ k g i j ,

where Υ i := Ω 1 i Ω .

References

Schouten tensor Wikipedia