I tried to define some methods, that construct for a given Transformation i:S->M the pulled back Vectorbundle i^*TM over S, which consists of Vectorfields along i, i.e. Y=X ° i, with X Vectorfield on M.
I have defined a method, TangentBundle(j,TjN) (wrong name, isn't it?), which takes a Transformation j from one frame into another and a specifier TjN and constructs a new bundle frame:
DGsetup([vars of Domain],[vars of Range],TjN)
Then, there is the method TangenVectors(j,TjN), which constructs the Pushforward(j,FrameBaseVectors of Domain) in TjN.
And finally, there is the method PullbackConnection(i,TjN,C), which takes again, a Transformation and the above constructed bundle, and should return the pulled back connection C in TjN. If C has christoffel symbols Gamma_ab^c, than the pulled back connection should have the symbols Gamma_ab^c j^a_k, there j^a_k is the jacobian of the transformation j. Unfortunately I get an error at the end of this method, which is:
Error, (in InfoProcTable["TensorDensityType"]) improper op or subscript selector
Please help me to figure out, what is going on. I think I could figure out the failure. I'm going to test this solution.
I have attached the file to this anwser: tangentbundle.mw