Timeline for A non-vanishing vector field on $S^3$ whose flow does not preserve any transversal foliation
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 6, 2019 at 14:05 | comment | added | Tom Goodwillie | Yes, I believe so. The Reeb foliation of $S^3$, in which there is one closed leaf, a torus, is invariant under such a flow. | |
| Jun 6, 2019 at 6:03 | comment | added | Ali Taghavi | In the other words, let Y be the unit norm vector foeld tangent to the hopf foberation. Is there a foliation which is invariant under flow of Y? | |
| Jun 6, 2019 at 6:00 | comment | added | Ali Taghavi | My apology for my several messages. In the example you provided, what about if we require the foliation would be flow invariant but not necessarilly transversal? | |
| Jun 5, 2019 at 18:51 | comment | added | Ali Taghavi | Here are some related results math.ucla.edu/~honda/confoliations.pdf | |
| Jun 5, 2019 at 12:10 | vote | accept | Ali Taghavi | ||
| Jun 5, 2019 at 12:10 | comment | added | Ali Taghavi | Let $X$ be a non vanishing vector field on $S^3$. Is there a 2 dimensional distribution transvers to $x$ which is invariant under $X-$flow? | |
| Jun 4, 2019 at 15:34 | comment | added | Ali Taghavi | Yes I see.In fact $\pi (L)$ is closed because if $L$ does not intersect a vertical circle then it does not intersect a tube neighborhood of that circle.(Because of foliation chart). | |
| Jun 4, 2019 at 12:18 | comment | added | Tom Goodwillie | If $F_2$ has, for example, only one closed leaf, then there cannot be a transverse vector field whose flow takes leaves to leaves. On the other hand, if $F_2$ is oriented then some transverse vector field (and 1 dimensional foliation) must exist. | |
| Jun 4, 2019 at 12:13 | comment | added | Tom Goodwillie | The fact that $\pi(L)$ is closed uses the compactness of the fibers of $\pi$. | |
| Jun 4, 2019 at 7:36 | comment | added | Ali Taghavi | According to the remaining part of my question, let we have a 1 dimensional foliation $F_1$ of $S^3$ which admit a 2 dimensional transversal foliation $F_2$. Is there a non vanishing vector field $X$ tangent to $F_1$ whose flow preserves the leaves of $F_2$. This property is frequently used in the Linked paper mentioned in this post:"Number theory and dynamical system on Foliated space" | |
| Jun 4, 2019 at 7:27 | comment | added | Ali Taghavi | I think your second paragraph is equivalent to the following 'a combination of Rank theorem and inverse function theorem, impliy that such section you mentioned exist", Yes? | |
| Jun 3, 2019 at 19:47 | comment | added | Ali Taghavi | Thank you for this very interesting answer. So i get the answer to my previous comment"Why $\pi :L \to S^2$ is surjective?" According to your answer, I realize that $\pi(L)$ is a clopen subset of $S^2$, hence is whole $S^2$. | |
| Jun 3, 2019 at 13:52 | history | edited | Tom Goodwillie | CC BY-SA 4.0 |
added 20 characters in body
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| Jun 3, 2019 at 12:47 | history | answered | Tom Goodwillie | CC BY-SA 4.0 |