Commuting Covariant Derivatives in Product Manifolds

[Lorenz Eberhardt]

Hi,

I am struggling to implement correctly a product manifold in xAct. My main problem is that I cannot get covariant derivatives of the different components of the product to commute. Here is a minimal example:


<< xAct`xTensor`
DefManifold[M1, 3, {\[Mu], \[Nu], \[Lambda], \[Rho], \[Sigma], \[Tau]}]
DefManifold[M2, 3, {a, b, c, d, e, f}]
DefManifold[M3, {M1, M2}, {M, P, Q, R, T}]
DefMetric[1, gM1[-\[Mu], -\[Nu]], CD1, SymbolOfCovD -> {";", "\[Del]"}]
DefMetric[1, gM2[-a, -b], CD2, SymbolOfCovD -> {";", "\[Del]"}]
SetOptions[ContractMetric, OverDerivatives -> True]
DefProductMetric[g[-M, -P], {{TangentM1, 1}, {TangentM2, 1}}, CD,  SymbolOfCovD -> {";", "\[Del]"}]
DefTensor[\[Phi][], M3]

Then neither

CD[\[Mu]]@CD[a]@\[Phi][] - CD[a]@CD[\[Mu]]@\[Phi][] // SortCovDs

nor

CD1[\[Mu]]@CD2[a]@\[Phi][] - CD2[a]@CD1[\[Mu]]@\[Phi][] // SortCovDs

simplify, also not upon applying ExpandProductMetric. Ideally, I would like that all covariant derivatives of M1 are to the left of those of M2. Is there any way to achieve this?

Any help would be greatly appreciated!

Best,

Lorenz