@ecterrab thanks a lot for your help. One last thing, how do I work in 3D? So I have a 3D metric and all these tensors will be computed in 3D?

@ecterrab thanks for your help.

I have done some research and I find using DifferentialGeometry better suits my needs than Physics.

Can I define a connection in terms of gamma coefficients? How do I assign values to these coefficients in DifferentialGeometry?

@ecterrab thanks again. I haven't done anything on Maple as I am coming from a different platform and trying to figure out how things work. What I want to do is compute the tensors in terms of the Weyl connection, which is defined differently than the Levi-Civita connection and hence the Christoffel symbols.

Can I define the connection then and that should result in new definitions for the Christoffel symbols? Or is the only solution to redefine everything?

@ecterrab thanks for your answer. I thought I wouldn't need to redefine everything: just Christoffel and Maple would use the default definitions for the tensors in terms of the new Christoffel. 
Anyway, how do I define Gamma with multiple values? An array?

P.S. It wasn't me who posted and deleted that question.