Timeline for A metric naturally arise from the Euclidean symplectic structure?
Current License: CC BY-SA 4.0
9 events
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| May 20, 2020 at 11:41 | comment | added | Lev Soukhanov | Considering similar theory for $1$-dimensional distributions they are always integrable, and so I think you can not define the similar theory for them. You can probably obtain the sequence of metrics by allowing to move transversely to the foliation but making it increasingly costly, but I'm not sure how useful this concept is. | |
| May 20, 2020 at 11:37 | comment | added | Lev Soukhanov | I think if you restrict codimension 1 distribution on a sphere you will obtain codimension 1 distribution. For example, consider $S^3$ embedded in $\mathbb{R}^4 = \mathbb{C}^2 = \mathbb{H}$. Then, the distribution $\text{Ker}(\omega(v, *))$ is also euclidean orthogonal to $I(v)$ due to the identity $\omega(v, x) = g(Iv, x)$. Now, it is actually generated by vector fields $Jv$ and $Kv$ ($I J K$ being quaternion imaginary units). Commutator of $J$ and $K$ is indeed $I$, so this distribution is non-integrable and generates the whole tangent space. | |
| May 18, 2020 at 18:04 | comment | added | Ali Taghavi | Note that non of the the three standard vector fields of S^3, which produce foliation by circles, ,are tangent to D. | |
| May 18, 2020 at 18:03 | comment | added | Ali Taghavi | thank you again for your answer. i confess that i did not read the book of Gromov yet. I just brows it but i will read it. But just a few more questions: Is the distribution I considered totally non integrable that is the liealgebra of vector field tangent to D is thewhole algebra of vector field? Moreover is there a 1 dimensional folition of S^3 tangent to D(The intersection of D with S^3) such that the foliation has no closed leaf? On the opposite extrem, is there a 1 dimensional foliation of 3 sphere tangent to D whose all leaves are closed? | |
| May 14, 2020 at 10:13 | history | edited | Lev Soukhanov | CC BY-SA 4.0 |
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| May 14, 2020 at 10:08 | comment | added | Lev Soukhanov | Considering this applied thematic the thing to google is "sub-riemannian model of visual cortex". | |
| May 13, 2020 at 21:16 | comment | added | Ali Taghavi | Thank you very much for your very interesting answer. | |
| May 13, 2020 at 21:15 | vote | accept | Ali Taghavi | ||
| May 13, 2020 at 20:43 | history | answered | Lev Soukhanov | CC BY-SA 4.0 |