Timeline for The integral of torsion
Current License: CC BY-SA 3.0
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| Jun 9, 2023 at 12:26 | answer | added | Tong | timeline score: 0 | |
| Jan 28, 2023 at 13:40 | comment | added | Ali Taghavi | @WlodAA Thank you very much for your very helpful comment | |
| Jan 28, 2023 at 13:34 | comment | added | Wlod AA | One needs to double check this information: I ran in 1974/5 into a pre-WWII paper in German authored by Karol Borsuk that stated and proved the mentioned theorem about the integral of torsion over a closed curve contained in $\ S^2\ $ being $\ 0$. | |
| Jan 28, 2023 at 12:41 | answer | added | Anton Petrunin | timeline score: 1 | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| May 23, 2016 at 10:07 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| S Apr 20, 2016 at 17:45 | history | bounty ended | CommunityBot | ||
| S Apr 20, 2016 at 17:45 | history | notice removed | CommunityBot | ||
| Apr 17, 2016 at 13:18 | comment | added | Andreas Cap | I just want to remark that this is a mix of intrinsic and extrinsic geometry. You have to view the sphere as a specific submanifold of $\mathbb R^3$, since the torsion of $\gamma$ refers to viewing $\gamma$ as a curve in $\mathbb R^3$. So a priori if you look for generalizations to other surfaces or to Riemannian geometry, you are looking for extrinsic invariants of a curve, whose integral vaishes provided that the curve lies in a given hypersurface. | |
| Apr 17, 2016 at 12:48 | comment | added | Sebastian Goette | @MikhailKatz Take a family of spheres with radius tending to $\infty$ that touch the plane in some point not too far away from the curve, now project onto each of these spheres. Then the integral condition is preserved by the quoted theorem, and the image curves approximate the original one. | |
| Apr 17, 2016 at 11:23 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Apr 14, 2016 at 12:30 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Apr 12, 2016 at 17:54 | comment | added | Ali Taghavi | for example the resulting integral would be some thing like $\epsilon+\delta$+higher order | |
| Apr 12, 2016 at 17:17 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Apr 12, 2016 at 16:47 | comment | added | Ali Taghavi | @MikhailKatz this is just a feeling not an absolute idea.For example imagine that we have a two parameter family of Jordan curves such that the torsion-integral is a function of these two parameters and some thing like implict function theorem can produce the curves with zero integral torsion. | |
| Apr 12, 2016 at 16:41 | comment | added | Mikhail Katz | How are you going to perturb a plane Jordan curve so as to keep the integral condition? | |
| S Apr 12, 2016 at 16:38 | history | bounty started | Ali Taghavi | ||
| S Apr 12, 2016 at 16:38 | history | notice added | Ali Taghavi | Authoritative reference needed | |
| S Dec 22, 2014 at 5:56 | history | bounty ended | CommunityBot | ||
| S Dec 22, 2014 at 5:56 | history | notice removed | CommunityBot | ||
| Dec 21, 2014 at 18:43 | comment | added | Ali Taghavi | @JoonasIlmavirta My deep thanks for your attention to my question and your valuable comment and references. | |
| Dec 21, 2014 at 17:32 | comment | added | Joonas Ilmavirta | Here are two references to get you started: arxiv.org/abs/1303.6114 and arxiv.org/abs/1410.2114 They both contain references to results in this direction. The first paper also discusses the case of manifolds with boundary, but this is somewhat different. The answer is known on some closed manifolds, like Lie groups and Anosov manifolds. (The problem of recovering a function from its integrals over lines or similar objects has been studied a lot in Euclidean spaces. Google "Radon transform" if you want to learn more. The literature on it is vast.) | |
| Dec 21, 2014 at 11:15 | comment | added | Ali Taghavi | @JoonasIlmavirta thank you very much for the comment. Could you please give some reference or examples about these? | |
| Dec 19, 2014 at 18:05 | comment | added | Joonas Ilmavirta | It is one of the main problems in integral geometry to find out which functions (or vector fields or tensor fields) on a Riemannian manifold integrate to zero over all closed geodesics. Results on this problem might not answer exactly the question you ask but they seem similar. | |
| Dec 19, 2014 at 17:54 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
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| S Dec 14, 2014 at 4:22 | history | bounty started | Ali Taghavi | ||
| S Dec 14, 2014 at 4:22 | history | notice added | Ali Taghavi | Authoritative reference needed | |
| Jun 28, 2014 at 20:07 | history | edited | Vidit Nanda | CC BY-SA 3.0 |
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| Jun 28, 2014 at 17:53 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jun 28, 2014 at 11:21 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jun 27, 2014 at 22:00 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jun 27, 2014 at 21:55 | history | asked | Ali Taghavi | CC BY-SA 3.0 |