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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 27th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexjust for completeness, am giving this its own little entry, finally <a href="https://ncatlab.org/nlab/revision/basic+differential+form/1">v1</a>, <a href="https://ncatlab.org/nlab/show/basic+differential+form">current</a>

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 27th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ### For general submersions 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. <a href="https://ncatlab.org/nlab/revision/diff/basic+differential+form/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/basic+differential+form/2">v2</a>, <a href="https://ncatlab.org/nlab/show/basic+differential+form">current</a>

   Added:

   For general submersions

   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.

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. We need to do something about the links to *[[contraction]]*... Let me see....

   Thanks.

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexFirst of all, I have now made the links for "contraction" point to *[[tensor contraction]]*. Next to improve the disambiguation at *[[contraction]]*... <a href="https://ncatlab.org/nlab/revision/diff/basic+differential+form/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/basic+differential+form/3">v3</a>, <a href="https://ncatlab.org/nlab/show/basic+differential+form">current</a>

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

   Next to improve the disambiguation at contraction…

   diff, v3, current