Covariant derivative for symbolic tensors

Question

I want to define a "prefix" (D_i) covariant derivative operator CD[] for symbolic tensors in form of a function, i.e. for

$Assumptions={Element[g,Arrays[{3,3},Reals,Symmetric[{1,2}]]]}

also

Element[CD[g],Arrays[{3,3,3},Reals,Symmetric[2,3]]]

should be made an assumption, an additional slot is created on the left. So far the only way I found to make this possible is defining a function

$Assumptions = {Element[v,Vectors[3, Reals]]}

CDdef[a_] := (AppendTo[$Assumptions, 
    Element[CD[a],Arrays[{3}~Join~TensorDimensions[a]],Reals, 
      TensorSymmetry[TensorProduct[v, a]]]];
      $Assumptions =  DeleteDuplicates[$Assumptions]; CD[a])

and effectively writing all the assumption by hand into $Assumptions. But this could generate a huge list in bigger calculations. All my attempts of the kind

$Assumptions={Element[CD[a_?SomeSymmetryQ],Arrays["one more slot"]]}

to use pattern that are at least valid for all tensors with the same symmetries have failed. Is there any effective way to proceed?

Many thanks, Hendrik

Answer 1

Good question; the notion of a tensorial (covariant) derivative is something that is missing in Mathematica AFAIK. I can think of two ways to proceed:

Option 1

One way is to overload the TensorRank, TensorDimensions, and TensorSymmetry functions for patterns that have head CD:

CD /: TensorRank[CD[tensor_]] := TensorRank[tensor] + 1

CD /: TensorDimensions[CD[tensor_]] := Join[{First[#]}, #]& @ TensorDimensions[tensor]

CD /: TensorSymmetry[CD[tensor_]] := TensorSymmetry[tensor] /.
  (h : (Cycles | Symmetric | Antisymmetric))[list_List] :> h[list + 1]

Note that this is a bit rough on the edges and gives incorrect results if tensor hasn't been defined as a tensor or is a scalar, but it shouldn't be too hard to improve the code.

The following then works as expected:

$Assumptions = m ∈ Matrices[{4, 4}, Reals, Symmetric[{1, 2}]];

TensorRank[CD@m]

3

TensorDimensions[CD@m]

{4, 4, 4}

TensorReduce@TensorTranspose[CD@m, {1, 3, 2}]

CD[m]

Option 2

However, if your covariant derivatives is commutative (like for instance the partial derivative), a better way would be to simply define it as a vector:

$Assumptions = CD ∈ Vectors[4, Reals] && m ∈ Matrices[{4, 4}, Reals, Symmetric[{1, 2}]]

If you like the CD[tensor] notation, you may even define

CD[tensor_] := TensorProduct[CD, tensor]

All the above commands then give the same result. In addition, multiple derivatives are symmetric:

TensorSymmetry[CD@CD@m]

{{Cycles[{{1, 2}}], 1}, {Cycles[{{3, 4}}], 1}}

So only use this if your derivatives actually commute!