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CurvatureRelationsBianchi

CurvatureRelationsBianchi[cd]
gives the contracted second Bianchi identities for the curvature tensors of the covariant derivative cd.
CurvatureRelationsBianchi[cd, Riemann]
gives only the identities for the Riemann tensor.
CurvatureRelationsBianchi[cd, Ricci]
gives only the identities for the Ricci tensor.
  • The second Bianchi identity for the Riemann tensor is ∇_([a)R_(bc]d)^e-T^f_([ab)R_(c]fd)^e=0, where T is the torsion.
  • The identities returned by CurvatureRelationsBianchi are the first contraction ∇_dR_(abc)^d= ∇_bR_(ac)-∇_aR_(bc)+ R_(adc)^eT^d_(be)- R_(bdc)^eT^d_(ae)- R_(dc)T^d_(ab) and the second contraction of the second Bianchi identity.
  • The second contraction is only defined when there is a metric.
  • Torsion is by default zero, but can be turned on with the option Torsion -> True of DefCovD.
  • CurvatureRelationsBianchi returns replacement rules.