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Added:
Given a submersion $p\colon\to B$, one may ask: which differential forms on $E$ are pullbacks of differential forms on $B$?
If the fibers of $p$ are connected (otherwise the characterization given below is valid only locally in $E$), the answer is provided by the notion of a basic form: a form $\omega$ is basic if the following two conditions are met:
The contraction of $\omega$ with any $p$-vertical vector field is zero.
The Lie derivative of $\omega$ with respect to any $p$-vertical vector field is zero.
Using Cartan’s magic formula, in the presence of the first condition, the second condition can be replaced by the following one:
Thanks.
We need to do something about the links to contraction… Let me see….
First of all, I have now made the links for “contraction” point to tensor contraction.
Next to improve the disambiguation at contraction…
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