First of all, I have now made the links for “contraction” point to *tensor contraction*.

Next to improve the disambiguation at *contraction*…

Thanks.

We need to do something about the links to *contraction*… Let me see….

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:

- The contraction of $d\omega$ (where $d$ is the de Rham differential) with any $p$-vertical vector field is zero.

just for completeness, am giving this its own little entry, finally

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