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
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| May 20, 2012 at 18:09 | comment | added | Robert Bryant | Let me point out that the overdetermined nature of the problem when $mk > m{+}k$ is not much evidence that it can't be solved. Usually, one worries about a system being overdetermined when one is trying to solve an initial value problem on a hypersurface. However, in the overdetermined cases here, one needs to solve a system in which one only specifies initial values along a submanifold of codimension $k>1$. It could happen (and I don't know whether it does) that the appropriate codimension for initial values for this system is $k$ or less, in which case, it might be solvable anyway. | |
| Apr 24, 2012 at 0:18 | comment | added | Miranda Holmes-Cerfon | sorry, I guess I misunderstood the purpose of that tool, I can reverse that. Thanks for the ideas though. | |
| Apr 23, 2012 at 20:59 | comment | added | Anton Petrunin | I am surprised that you mark my answer as accepted; I did not give a cmplete solution even in the overdetermined case. | |
| Apr 23, 2012 at 16:01 | vote | accept | Miranda Holmes-Cerfon | ||
| Apr 23, 2012 at 23:13 | |||||
| Apr 23, 2012 at 4:14 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Apr 23, 2012 at 1:45 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Apr 22, 2012 at 22:12 | history | answered | Anton Petrunin | CC BY-SA 3.0 |