Timeline for twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 30, 2012 at 19:45 | vote | accept | issoroloap | ||
| Dec 30, 2012 at 19:44 | vote | accept | issoroloap | ||
| Dec 30, 2012 at 19:44 | |||||
| Dec 30, 2012 at 19:43 | comment | added | issoroloap | Yes i meant precisely that. I'll think about it a little more, but this is already more than satisfactory. Thanks again. | |
| Dec 30, 2012 at 18:47 | comment | added | Robert Bryant | @issoroloap: $D_r$ does not have linearity properties in any usual sense because the fibers of the bundle $\otimes^2_r(TM)$ don't have a linear structure. I guess you could ask what you could say if you had a linear combination $\Phi=a\phi+b\varphi$ with the property that this was a section of $\otimes^2_r(TM)$ for all $a$ and $b$ (or an open subset of such coefficients). Offhand, I don't know of any examples of this that don't keep the two bundles $\lambda_\Phi$ and $\rho_\Phi$ fixed (in which case, the 'superposition formula' is kind of trivial), but I imagine that there could be some. | |
| Dec 30, 2012 at 18:26 | comment | added | issoroloap | let me elaborate: one cool thing about the equation [\Pi,\Pi]=\Pi^\sharp(\phi) for the integrability of antisymmetric case is that one can put $\Pi=\Pi_0+\epsilon \Pi_1$ and get a compatibility condition between $\Pi_0$ and $\Pi_1$ in order for $\Pi$ to generate an integrable distribution (this is the idea of bihamiltonian systems). In the fully Poisson case it reduces to $[\Pi_0,\Pi_1]=0$ of course, but if we start with a Poisson tensor $\Pi_0$ but perturb it to anything integrable (but still antisymmetric) we get a nice and explicit system of equations involving $\Pi_0$, $\Pi_1, \phi$. | |
| Dec 30, 2012 at 18:04 | comment | added | issoroloap | great, thank you Robert! one question though: what about the linearity properties of you differential operator? I see that it is homogeneous of dregree r+2 with respect to multiplication of $\phi$ by a constant (even a function). what about linear combinations of tensors $a\phi+b\varphi$? | |
| Dec 30, 2012 at 15:45 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added information about the general case
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| Dec 30, 2012 at 14:52 | comment | added | issoroloap | Thank you also for this reply. This is an extremely direct test indeed. | |
| Dec 29, 2012 at 14:50 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added information about the almost Poisson case
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| Dec 29, 2012 at 0:08 | history | edited | Robert Bryant | CC BY-SA 3.0 |
removed the antisymmetric case for more work
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| Dec 28, 2012 at 23:25 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed typos and normalizations
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| Dec 28, 2012 at 23:03 | history | answered | Robert Bryant | CC BY-SA 3.0 |