Timeline for Connectedness of the symplectic automorphism of the 2-sphere $S^2$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 27, 2014 at 17:13 | comment | added | Oldřich Spáčil | @Entaou Yes, time-dependent functions, that's correct and that's why I wrote loosely speaking ;-) | |
| Oct 27, 2014 at 13:21 | comment | added | Entaou | @Oldřich Spáčil Thank you! I think that your exact meaning is that: Ham(M,ω) is generated by time-dependent functions. | |
| Oct 25, 2014 at 21:17 | comment | added | Oldřich Spáčil | @Entaou More generally, I understand that you are interested in the connected component of the identity in the group of symplectic diffeomorphisms. In the case of simply connected manifolds this connected component is exactly the group $\mathrm{Ham}(M, \omega)$ of Hamiltonian diffeomorphisms of $(M, \omega)$ -- these are the diffeomorphisms "generated by functions, indeed" (loosely speaking). And, as far as I can remember, all coadjoint orbits are simply connected. | |
| Oct 25, 2014 at 21:10 | comment | added | Oldřich Spáčil | @Entaou A symplectic form on a 2-dim manifold is the same as an area form, so the group of symplectomorphisms is the same as the group of area preserving diffeomorphisms (and preserving the orientation). In the case of the 2-sphere, the group $\mathrm{Diff}(S^{2}, \omega)$ is homotopy equivalent to $\mathrm{SO}(3)$ by Smale. In particular, it is connected. | |
| Oct 23, 2014 at 17:23 | comment | added | Ryan Budney | Close to a duplicate: mathoverflow.net/questions/9335/… | |
| Oct 23, 2014 at 16:19 | history | asked | Entaou | CC BY-SA 3.0 |