I Pullback of a vector field in relativity

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)?


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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.
Just wanted to make sure, thanks!

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