Timeline for Integrability of certain distribution associated to a connection form on the total space of a principal bundle (Principal Frobenius condition)
Current License: CC BY-SA 4.0
6 events
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| Aug 30, 2023 at 5:49 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing. A note: hyperlinking works also across different posts of the same thread, so `\eqref{1}` gives \eqref{1} in any post of the same Q&A.
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| Jul 3, 2019 at 6:32 | comment | added | Ali Taghavi | My apology for asking this question again: According to your answer, does the conclusion in my previous comment, true? | |
| Jun 29, 2019 at 19:45 | comment | added | Ali Taghavi | So can we conclude from your answer that if we have a G principal bundle such that $\mathfrak{g}$ has trivial center then every arbitrary connection is flat since it is integrable? Is it realy the case? | |
| Jun 29, 2019 at 15:54 | comment | added | Tsemo Aristide | A proof of the Frobenius theorem works like so, let $X_1,..,X_p$ be vector fields in involution, one creates $X'_1,...,X'_p$ from $X_1,..,X_p$ which generates the same distribution such that $[X`_i,X'_j]=0$, then the flow of $X'_i$ define the foliation, so the involution in Frobenius theorem needs to be true for only one set of vector fields in involution. | |
| Jun 29, 2019 at 15:44 | comment | added | Ali Taghavi | Fundamental vector fields are vertical vector fields. But to check Frobenius condition we need consider arbitrary vector fields in our distribution. Right? | |
| Jun 29, 2019 at 14:37 | history | answered | Tsemo Aristide | CC BY-SA 4.0 |