Timeline for What are the minimal local models for Riemannian manifolds? A local question about isometric embeddings
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 30, 2017 at 9:23 | vote | accept | Saal Hardali | ||
| Oct 30, 2017 at 1:25 | comment | added | Deane Yang | Oops. Thanks for posting the correct link. | |
| Oct 30, 2017 at 1:18 | comment | added | j.c. | @DeaneYang Thanks for the comments! Here's a non-proxified link to Gromov's paper: ams.org/journals/bull/2017-54-02/S0273-0979-2016-01551-5 | |
| Oct 30, 2017 at 0:25 | comment | added | Deane Yang | Gromov proves a number of theorems about the existence of local isometric embeddings in his book Partial Differential Relations. He also has a recent survey paper (an outgrowth of his talk at the Nash memorial conference) in the Bulletin of the AMS. ams.org.proxy.library.nyu.edu/journals/bull/2017-54-02/… | |
| Oct 30, 2017 at 0:21 | comment | added | Deane Yang | A couple of quick comments: 1) The Janet-Cartan theorem is indeed the best possible because it is possible to construct a Riemannian metric for which there is not even a formal solution to the isometric embedding equation at a point. 2) As for extending the Janet-Cartan theorem to smooth metrics, the problem is that the system of PDEs is a really nasty one, unless suitable assumptions are made. Even when $n=2$ there is no complete solution. In higher dimensions there is very little known. I can only speculate that an approach that does not use standard PDE theory has to be used. | |
| Oct 29, 2017 at 22:08 | history | answered | j.c. | CC BY-SA 3.0 |