Let nabla be a connection on the tangent bundle of a manifold M and let chi(M) be the module of all vector fields on M.  The curvature tensor of nabla is the type (1,3) tensor R (contravariant rank 1, covariant rank 3) which, when viewed as a linear map R: chi(M) x chi(M) -> Hom(chi(M)), is given by

Let nabla be a connection on a vector bundle E -> M and let Sigma(E) be the module of all sections of E.  The curvature tensor of nabla is the type (3, 1) tensor R (contravariant rank 1, covariant rank 3) which, when viewed as a linear map R: chi(M) x chi(M) -> Hom(Sigma(E)) is given by R(X, Y)(Z) = nabla_X(nabla_Y(Z)) - nabla_Y(nabla_X(Z)) - nabla_[X, Y](Z).  Here X and Y are vector fields on M and Z is a section of E.

The first Bianchi identity for the curvature tensor of a connection on the tangent bundle of a manifold asserts that R + nabla(S) + S . S (skew-symmetrized on indices 1, 3, 4) = 0.  The second Bianchi identity asserts that nabla(R) + R . S (skew-symmetrized on indices 3,  4,  5) = 0.  Here S is the torsion tensor of the connection, nabla(S) the covariant derivative of S, S . S is the contract of S &t S over the 3rd and 5th indices, and R . S denotes the contraction of R &t S over the 3rd and 6th indices.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CurvatureTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-CurvatureTensor.

Bianchi1 := proc(R, C)

end:

Bianchi2 := proc(R, C)

end: