The Invar tensor package: Differential invariants of Riemann

The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6 . 10(23) objects with up to 12 derivatives of the metric. This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann. We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis. The process is as follows: (I) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6 . 10(5) relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4. and finally (6) simplifies invariants that can be expressed as product of dual invariants. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica) and Canon (for Maple).