nForum - Discussion Feed (basic differential form) 2023-06-03T07:48:59+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "basic differential form" (109003) https://nforum.ncatlab.org/discussion/10078/?Focus=109003#Comment_109003 2023-04-27T15:11:12+00:00 2023-06-03T07:48:59+00:00 Urs https://nforum.ncatlab.org/account/4/ 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

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

Next to improve the disambiguation at contraction

]]>
Urs comments on "basic differential form" (109002) https://nforum.ncatlab.org/discussion/10078/?Focus=109002#Comment_109002 2023-04-27T15:07:19+00:00 2023-06-03T07:48:59+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks. 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….

]]>
Dmitri Pavlov comments on "basic differential form" (109001) https://nforum.ncatlab.org/discussion/10078/?Focus=109001#Comment_109001 2023-04-27T14:57:58+00:00 2023-06-03T07:48:59+00:00 Dmitri Pavlov https://nforum.ncatlab.org/account/356/ Added: For general submersions Given a submersion p&colon;&rightarrow;Bp\colon\to B, one may ask: which differential forms on EE are pullbacks of differential forms on BB? If the fibers of ...

### 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.
]]>
Urs comments on "basic differential form" (78808) https://nforum.ncatlab.org/discussion/10078/?Focus=78808#Comment_78808 2019-06-27T16:25:01+00:00 2023-06-03T07:48:59+00:00 Urs https://nforum.ncatlab.org/account/4/ just for completeness, am giving this its own little entry, finally v1, current

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

]]>