Timeline for Legendrian Tubular Neighborhood Theorem
Current License: CC BY-SA 3.0
9 events
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| May 5, 2021 at 19:07 | history | edited | John Pardon |
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| Sep 10, 2013 at 20:58 | comment | added | Oldřich Spáčil | Yes, I do understand you're interested in the Lag -> Leg direction... In my second comment I just tried to point out that the other direction seems easier to me, especially if in your setup you start with a Legendrian submanifold and only then consider symplectizations etc. Otherwise I can't come up with anything better than Alan Weinstein's remark on the $\mathbb{R}$-equivariancy. | |
| Sep 10, 2013 at 20:10 | comment | added | Chris Gerig | Sorry, my question is focusing on the possibility to pass from Lagrangian neighborhood to Legendrian neighborhood (whereas the answers thus far are only showing me how to get a Legendrian neighborhood). | |
| Sep 10, 2013 at 19:05 | history | edited | Chris Gerig | CC BY-SA 3.0 |
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| Sep 10, 2013 at 17:23 | comment | added | Oldřich Spáčil | Also, my feeling is that you'd rather prove the Lagrangian nbhd theorem from the Legendrian one since you supposedly start with a contact manifold and its Legendrian and then consider the symplectization? If I'm not mistaken, the symplectization of $J^{1}K$ is symplectomorphic to $T^{*}(K\times \mathbb{R})$. | |
| Sep 10, 2013 at 17:13 | comment | added | Oldřich Spáčil | You're right that Moser's method works well for proving a Legendrian nbhd theorem -- a tubular neighbourhood of $K$ is contactomorphic to a neighbourhood of the zero section of the 1-jet bundle $J^{1}K$ as described in the answers below. A proof can be found in Geiges's book An Introduction to contact topology, Theorem 2.5.8 and Example 2.5.11. You can even make the contactomorphism to be strict, i.e. preserving a given contact form -- a proof of this version is in Kriegl, Michor: The convenient setting of global analysis, Proposition 43.18 (this book is available from Michor's website). | |
| Sep 10, 2013 at 9:24 | answer | added | Andy McI | timeline score: 1 | |
| Sep 10, 2013 at 6:55 | answer | added | Giovanni Moreno | timeline score: 1 | |
| Sep 10, 2013 at 0:03 | history | asked | Chris Gerig | CC BY-SA 3.0 |