Timeline for Is the action of diffeomorphisms transitive on the set of vector fields with prescribed zero locus?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 10, 2017 at 19:10 | comment | added | Ali Taghavi | @Qfwfq Aside from closed orbit or singularity, one can introduce two vector field on the plane which does not have singularity but they are not smooth related. For example : $X= \frac{\partial}{\partial x}$ and $Y=cos y \frac{\partial}{\partial x} +sin y \frac{\partial}{\partial y}$. The first vector field admits a global tranversal section but the second one does not. | |
| Jul 9, 2017 at 14:55 | vote | accept | Qfwfq | ||
| Jul 9, 2017 at 12:37 | answer | added | Thomas Rot | timeline score: 6 | |
| Jul 9, 2017 at 11:38 | comment | added | Qfwfq | @AndreasCap: yes. I should probably have written "the vanishing scheme" instead of "the vanishing set" (Though of course this would not be enough for a positive answer to my question, due to the comments above). | |
| Jul 9, 2017 at 8:33 | comment | added | Andreas Cap | The order of vanishing in an isolated zero is another additional invariant which may prevent vector fields from being related by a diffeomorphism. Think about $M=\mathbb R^n$, $V=\{0\}$ and the fields $|x|^{2k}\tfrac{\partial}{\partial x_1}$ for different $k$. | |
| Jul 8, 2017 at 18:05 | comment | added | Qfwfq | @ThomasRot: Oh yes, I see. Your two comments are two distinct ways in which things can go wrong. | |
| Jul 8, 2017 at 18:02 | comment | added | Thomas Rot | Another example is the hopf fibration on The threesphere, and a flow on the threesphere without equilibria where not all orbits are periodic | |
| Jul 8, 2017 at 18:00 | comment | added | Thomas Rot | No this is not true. Imagine the gradient flow of the height function on the circle. And a flow with two equilibria that always points counterclockwise. | |
| Jul 8, 2017 at 17:49 | history | asked | Qfwfq | CC BY-SA 3.0 |