Timeline for Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Dec 6, 2023 at 22:20 | vote | accept | Bilateral | ||
| Dec 3, 2023 at 18:58 | history | edited | Bilateral | CC BY-SA 4.0 |
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| Dec 3, 2023 at 18:50 | answer | added | Robert Bryant | timeline score: 1 | |
| Dec 2, 2023 at 19:14 | answer | added | Robert Bryant | timeline score: 3 | |
| Dec 1, 2023 at 18:42 | history | edited | Bilateral | CC BY-SA 4.0 |
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| Dec 1, 2023 at 18:42 | comment | added | Bilateral | Thanks for your answer. It is for some given pair, not any pair. If I am not mistaken, intuitively speaking $\Thetaî_{jk}$ should be "fixed" as prescribed by the Levi-Civita connection through Cartan's first structure equation associated with $h$. In more general terms, I fail to see what type of answer or information the Cartan-Kähler theory could given for this problem. | |
| Dec 1, 2023 at 18:14 | comment | added | Robert Bryant | Your phrase 'with $i_0$ and $j_0$ fixed' seems ambiguous to me. Do you mean 'for some given pair $i_0$ and $j_0$' or do you mean 'for any given pair $i_0$ and $j_0$'? In any case, it's not hard to give examples of functions $\Theta^i_{jk}$ for which there is no solution to the first system at all, even locally. | |
| Dec 1, 2023 at 17:51 | history | edited | Bilateral | CC BY-SA 4.0 |
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| Dec 1, 2023 at 17:49 | comment | added | Bilateral | Sorry, there was a typo, the condition on "$\sum_k^3 L_k e^k$" is that it is exact. In fact, we can assume that $L_k = \Theta^{i_0}_{j_0 k}$ where $i_0$ and $j_0$ are fixed. | |
| Dec 1, 2023 at 17:44 | history | edited | Bilateral | CC BY-SA 4.0 |
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| Dec 1, 2023 at 17:43 | comment | added | Bilateral | Thanks a lot for your answer. The unknown is the coframe $(e^1,e^2,e^3)$ with both $\Theta$ and $L$ as given data and fixed. What I don't understand is "The Cartan-Kaehler analysis is trivial if all $L_k$ are zero". Since the $\Theta^i_{jk}$ are fixed, how can all coframes be solutions? Don't the $\Theta^i_{jk}$ correspond to the Levi-Civita connection of the corresponding metric $h$? | |
| Dec 1, 2023 at 10:59 | comment | added | Ben McKay | The Cartan-Kaehler analysis is trivial if all $L_k$ are zero, clearly, since then all choices of framings $e^k$ are solutions. Do you have any further information about these $L_k$? | |
| Dec 1, 2023 at 10:37 | comment | added | Ben McKay | The unknown here is the coframing, I assume. Is that right? | |
| Dec 1, 2023 at 10:36 | comment | added | Ben McKay | Your equation should be $de^i=$ not $e^i=$. | |
| Dec 1, 2023 at 10:35 | comment | added | Ben McKay | Your following condition is that $\sum L_ke^k$ is closed? | |
| Dec 1, 2023 at 6:52 | history | asked | Bilateral | CC BY-SA 4.0 |