Timeline for An analytic vector field on $S^3$ whose all orbits are dense (à la Seifert conjecture, 2)
Current License: CC BY-SA 4.0
15 events
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| Jul 6, 2019 at 15:01 | comment | added | Ali Taghavi | @abx yes exactly. He consider the "revetment" de unit tangent bundle of $S^2$. | |
| Jul 6, 2019 at 14:21 | comment | added | abx | OK, you are right. The vector field is actually on $SO(3)$ (= the unitary tangent bundle to $S^2$), hence it lifts to $S^3$. | |
| Jul 6, 2019 at 14:07 | comment | added | Ali Taghavi | @abx in that paragraph of the paper of Ghys, however he starts the paragraph with 2 dimensional sphere but during the remaining part of the paragraph he is considering 3- sphere.Right? | |
| Jul 6, 2019 at 14:03 | comment | added | Ali Taghavi | @abx By Poincare -Bendixon theorem(or some similar methods) one can easilly shows that $S^2$ does not admit any vector field with a dense orbit. | |
| Jul 6, 2019 at 13:42 | comment | added | abx | The vector field mentioned in Ghys' paper is on $S^2$, not on $S^3$. | |
| Jul 6, 2019 at 12:41 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jul 6, 2019 at 11:31 | comment | added | Ali Taghavi | @Misha Yes but that counter example does not satisfy the condition of this question: the condition is: all orbits would be dense. | |
| Jul 6, 2019 at 10:28 | comment | added | abx | @Ali Taghavi: the difference is that $S^3=SU(2)$ is semi-simple. Every element is contained in a circle. | |
| Jul 6, 2019 at 9:42 | comment | added | Misha | Real-analytic counterexamples to Seifert conjecture were published by Kuperberg and Kuperberg in Annals of Math., 1996. | |
| Jul 6, 2019 at 8:54 | comment | added | Ali Taghavi | @abx Kronecker foliation is invariant in 2 torus. So in S^3 case os not possible that an orbit would be non closed? | |
| Jul 6, 2019 at 7:54 | comment | added | LeechLattice | $SO(4)=SO(3) ⊕ SO(3)$, and pick two incommensurable circle actions. | |
| Jul 6, 2019 at 7:39 | history | edited | YCor | CC BY-SA 4.0 |
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| Jul 6, 2019 at 7:29 | comment | added | abx | Every orbit of a left invariant vector field is closed (it is contained in the translate of a torus). | |
| Jul 5, 2019 at 20:22 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jul 5, 2019 at 20:13 | history | asked | Ali Taghavi | CC BY-SA 4.0 |