Timeline for Characterizing maximal powers of closed 2-forms in odd-dimensional manifolds
Current License: CC BY-SA 3.0
11 events
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| Jan 7, 2013 at 13:06 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed more typos and improved the exposition
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| Jan 1, 2013 at 17:03 | comment | added | Robert Bryant | @alvarezpaiva: Offhand, I don't remember where to find it mentioned in the literature, but, this kind of 'global' obstruction is known to occur. For example, if one takes the oriented path geometry on an oriented Riemannian surface that is given by the oriented curves of constant geodesic curvature $\kappa>0$, this admits an $\Omega$ ($=\omega$ since $n=1$) that is closed and nonvanishing, but there need not exist a Lagrangian $\lambda$ such that $\Omega = d(\delta\lambda)$, even if $\Omega$ is exact (which is not always the case either). (Examples: The flat torus and the round $2$-sphere.) | |
| Jan 1, 2013 at 16:02 | comment | added | alvarezpaiva | @Robert: it bothers me a bit that I don't see anywhere in the references I've looked at the condition that the pushforward (fiber integration) of $\Omega$ onto $M$ must be zero, which is the integral geometric kernel of the problem. Is this mentioned anywhere? | |
| Jan 1, 2013 at 15:12 | comment | added | Robert Bryant | @alavarezpaiva: Happy New Year to you, too! The 'microlocality' of the solutions is, as you say, a (known) problem, but you would have run into this problem immediately anyway because constructing the closed $2n$-form $\Omega$ is still nontrivial (in fact, when $n=1$, this is the heart of the problem) if you want to get 'local' solutions defined on $\pi^{-1}(U)$ for open sets $U\subset M$. However, when $n>1$, you'll find that 'most' oriented path geometries $E$ that are variational have only one Lagrangian, even locally (up to divergence equivalence), so this microlocal problem doesn't arise. | |
| Jan 1, 2013 at 11:19 | comment | added | alvarezpaiva | I'd be interested in a selected bibliography ! | |
| Jan 1, 2013 at 10:34 | comment | added | alvarezpaiva | Thanks Robert (and happy new year by the way) for this detailed answer. It was nice you provided the background. I was, as you guessed, "swiming back upstream" from $E$ to $\lambda$ and examining each condition on its own. The problem I have with the more practical method you mention is that it seems to yield Lagrangians that are only defined locally in the tangent ray space: locally in position and locally in direction. I would like to get solutions that are ony local in position: defined on sets of the form $\pi^{-1}(U)$, where $U$ is an open set in $M$. | |
| Jan 1, 2013 at 10:12 | vote | accept | alvarezpaiva | ||
| Jan 1, 2013 at 8:20 | history | edited | Robert Bryant | CC BY-SA 3.0 |
corrected some more typos and grammar
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| Jan 1, 2013 at 1:30 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed a typo
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| Jan 1, 2013 at 1:03 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed grammar and clarified statements
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| Jan 1, 2013 at 0:41 | history | answered | Robert Bryant | CC BY-SA 3.0 |