Timeline for Is there a parameterization of a neighbourhood of $x\in\mathbb{R}^n$ into two mutually orthogonal sets of variables, with one set parameterizing a pre-defined (n-k)-dimensional submanifold containing $x$?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 23, 2013 at 16:06 | answer | added | Robert Bryant | timeline score: 11 | |
| Apr 23, 2012 at 16:01 | vote | accept | Miranda Holmes-Cerfon | ||
| Apr 23, 2012 at 23:13 | |||||
| Apr 22, 2012 at 22:16 | comment | added | Anton Petrunin | @Ryan, tubular neighbourhood is not good here; in this case the equitation holds only on $M$, but not in a neighborhood. | |
| Apr 22, 2012 at 22:12 | answer | added | Anton Petrunin | timeline score: 6 | |
| Apr 22, 2012 at 17:57 | comment | added | Ryan Budney | Yes, of course you can. This comes up in the differential-geometric proof of the tubular neighbourhood theorem, for example. You can write out them map fairly explicitly in terms of holonomy, or in your case using your gradient vectors. | |
| Apr 22, 2012 at 15:46 | comment | added | Yuchen Liu | You can extend the normal vectors of $M$ in small scale and by inverse function theorem they do not intersect with others in a small neighborhood of $M$, which is isomorphic to a neighborhood of the zero section of the normal bundle of $M$. | |
| Apr 22, 2012 at 15:29 | history | asked | Miranda Holmes-Cerfon | CC BY-SA 3.0 |