I Pullback of a vector field in relativity

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Summary
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

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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.
 
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Just wanted to make sure, thanks!
 

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