Tensor[SchoutenTensor] - calculate the Schouten tensor of a metric
Calling Sequences
SchoutenTensor(g)
SchoutenTensor(g, R)
Parameters
g - a metric tensor on the tangent bundle of a manifold
R - (optional) the curvature tensor of g
Description
Examples
Let gab be metric (of any signature) on the tangent bundle of a manifold M of dimensionn>2. Let the Ricci tensor of g be Rab with scalar curvature R=gabRab. The the Schouten tensor Sab of gab is the symmetric tensor
Sab=1n−2Rab−R2n−1gab.
The first calling sequence computes the curvature tensor, Ricci tensor, and Ricci Scalar directly from the given metric. The second calling sequence uses the given curvature tensor Ra bcd to compute the Ricci tensor via Rbd=Ra bad and then computes the scalar curvature using the given metric via R =gabRab.
This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SchoutenTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-ProjectiveCurvature.
withDifferentialGeometry:withTensor:
Example 1.
Calculate the Schouten tensor of a metric.
DGsetupx,y,z,P
frame name: P
g≔evalDGexpλxdx&tdx+dy&tdy+dz&tdz
g:=ⅇλxdxdx+ⅇλxdydy+ⅇλxdzdz
SchoutenTensorg
18λ2dxdx−18λ2dydy−18λ2dzdz
Example 2.
Calculate the Schouten tensor from a metric and curvature tensor.
R≔CurvatureTensorg
R:=−14λ2D_ydzdydz+14λ2D_ydzdzdy+14λ2D_zdydydz−14λ2D_zdydzdy
SchoutenTensorg,R
See Also
DifferentialGeometry
CurvatureTensor
RicciTensor
RicciScalar
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