Pullback of Vector Field in Relativity: Restrictions?

[Pencilvester]

TL;DR

Shouldn’t the pullback of a vector (and pushforward of scalar and form, etc.) be easily definable for coordinate transformations?

Since coordinate transformations should be one-to-one and therefore invertible, wouldn’t there be no restriction on pushforwarding or pullbacking whatever fields we feel like (within the context of coordinate transformations)?

 

[Orodruin]

Pullbacks and pushforwards in differential geometry deal with maps between manifolds, not coordinate transformations.

That being said, if you have a smooth map from a manifold onto itself (such as defined by the flow of a vector field), then you can make pushforwards in either direction based on the function or its inverse. This is necessary for example for Lie derivatives of general tensors.

 

[Pencilvester]

Just wanted to make sure, thanks!