Timeline for Can a Lagrangian submanifold of ${\mathbb R}^{2n}$ be dense ($n>1$)?
Current License: CC BY-SA 3.0
10 events
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| Jan 22, 2012 at 3:53 | comment | added | Richard Montgomery | @Robert. I don't know yet. I will try to take time this week to find out. The dichotomy between this Panov result on an unoriented line field with singularities on the plane , and the Poincare-Bendixson theorem for an oriented $C^1$ line field with singularities (a.ka. vector field) is pretty dramatic, so seems worth looking into. | |
| Jan 19, 2012 at 12:34 | comment | added | Robert Bryant | @Richard: Thanks. I have a question about the $n=1$ example you cited. I looked at the paper and got a general idea of how the construction goes, but it wasn't clear to me (since I haven't had time to study the details) whether this example of a dense, connected $1$-dimensional submanifold in $\mathbb{R}^2$ is real-analytic. Is it? The dense real-analytic curves that I describe in my answer are not submanifolds, of course. | |
| Jan 18, 2012 at 14:33 | comment | added | Richard Montgomery | You the man, Robert! Very clear. And the concrete example of the dense curve is a big help. -thank you. | |
| Jan 18, 2012 at 14:29 | vote | accept | Richard Montgomery | ||
| Jan 17, 2012 at 18:56 | history | edited | Robert Bryant | CC BY-SA 3.0 |
corrected 'embedded' to 'submanifold'
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| Jan 17, 2012 at 18:22 | comment | added | Liviu Nicolaescu | @ Robert: Very nice examples. | |
| Jan 17, 2012 at 16:12 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added information about embeddings and the n=1 case
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| Jan 17, 2012 at 15:50 | comment | added | Robert Bryant | @Liviu: Yes, you are right, the $n$ even case is clear for this reason, but, somehow, it seems like cheating. Moreover, one could get examples for all $n$ by starting with a dense (analytic, if you like) curve in $\mathbb{R^2}$ and taking the $n$-fold product of it in $\mathbb{R}^{2n}$ (as BS pointed out in the comments to the question). It seems that there should be some construction that doesn't depend on complex analysis or products. | |
| Jan 17, 2012 at 15:01 | comment | added | Liviu Nicolaescu | @ Robert You can take direct products of your example to produce higher dimensional examples, $n$ even. | |
| Jan 17, 2012 at 12:38 | history | answered | Robert Bryant | CC BY-SA 3.0 |