How to define two distinct partial derivative operators on two different 1-dimensional submanifolds?

[Behnoush Khavari]

Hi,

Thanks for this nice forum.

I have encountered a problem while trying to split a 4d manifold into three submanifolds, two of which are 1-dimentional, and hence, their corresponding covariant derivative being just partial derivative.

The problem arises in expressions in which two partial derivatives are applied on a term, each of which corresponds to one of the two distinct 1-dimensional submanifolds. When I apply the command 

ToCanonical[#, UseMetricOnVBundle -> None] & , on this type of two-derivative terms, indices related to the two distinct manifolds are contracted in a wrong way, giving rise to a wrong result.

I have written the problem in a short mathematica notebook and appreciate it very much if someone could help me with it.

Thank you,

Behnoush