Timeline for Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 25, 2018 at 23:29 | comment | added | Ali Taghavi | Of course your answer and the linked paper is interesting. Thanks for your attention to my question. You answer is not useless as i gave my +1. | |
| Sep 25, 2018 at 16:12 | comment | added | L.F. Cavenaghi | @AliTaghavi, I see the problem and I agree with you that this is not trivial. Have you seen the paper I've sent to you? Perhaps one can obtain your desired result from there. I am so sorry for my naive approach, although it was fruithful to realize some properties for the immersion (as the equation I've stated). I agree that perhaps this is uselss to you, but who knows if it cannot be useful sometime? | |
| Sep 25, 2018 at 7:40 | comment | added | Ali Taghavi | @L.F.Cavenaghi The point is that the natural symplectic structure comes from the cotangent bundle. On the other hand when $N$ is a submanifold of $M$, there is no a canonical embedding of $T^* N$ in $T^*M$ such that the embedding would be independemt of any metric. | |
| Sep 25, 2018 at 7:36 | comment | added | Ali Taghavi | @L.F.Cavenaghi Thank you for your comment. The linked paper is a very interesting paper. | |
| Sep 24, 2018 at 20:51 | comment | added | L.F. Cavenaghi | You are both right, I misread the questions at first, so exactness is not a problem, but I endorse I am pretty sure that this is a not trivial question, this is why I have suggested one approach to at least look at an immersion equation. Also, on the paper I've sent you it discuss the problem on a more general frame, considering embbedings of symplectic space in to other symplectic spaces, not necessarily $\mathbb{R}^N.$ | |
| Sep 24, 2018 at 19:33 | comment | added | Ali Taghavi | @FrancoisZiegler Yes the 2 forms on TM are exact. So there is no matter of exactness. The matter is whether the "Naturality" mentioned by Paul is realy the case? What do you think about that?To be honnest I think that the problem is nontrivial. | |
| Sep 24, 2018 at 9:11 | comment | added | Ali Taghavi | Thank you for introducing me the linked paper. i will read your answer as soon as posible. | |
| Sep 24, 2018 at 3:52 | comment | added | Francois Ziegler | OP’s 2-form $\omega_g$ is on $TM$ (not $M$) and is always exact. | |
| Sep 24, 2018 at 0:34 | history | edited | L.F. Cavenaghi | CC BY-SA 4.0 |
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| Sep 24, 2018 at 0:28 | history | answered | L.F. Cavenaghi | CC BY-SA 4.0 |