Timeline for Solutions of equations characterizing a complex structure
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Mar 31, 2016 at 12:40 | comment | added | Amir Baghban | @RobertBryant Dear Prof. Bryant, I get more than what I expected and I am grateful for your complete answers and explanations. | |
| Mar 31, 2016 at 10:37 | vote | accept | Amir Baghban | ||
| Feb 20, 2016 at 12:25 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Rearranged the question a bit for clarity and fixed the incorrect equations specifying integrability.
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| Feb 20, 2016 at 4:11 | comment | added | Robert Bryant | @Solar: You have improved your question because now you are correctly stating that the number of (real) equations on $z$ is $2n$ instead of $3n$, but you still don't have them right because, for example, $z = i$ satisfies your equations, but the corresponding almost complex structure $J_{1,0}$ is not integrable. You seem to be having problems working things out in coordinates. Perhaps you are confusing projective coordinates (which carry geodesics to straight lines) with conformal coordinates. Would you like me to try editing it? | |
| S Feb 19, 2016 at 16:09 | history | suggested | Amir Baghban | CC BY-SA 3.0 |
According to the Prof. Bryant's explanations in a mail, the equations were wrong and I changed them to a right equations.
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| Feb 19, 2016 at 15:52 | review | Suggested edits | |||
| S Feb 19, 2016 at 16:09 | |||||
| Feb 16, 2016 at 13:15 | comment | added | Robert Bryant | @Solar: By the way, I just realized and feel that I should point out that there is something wrong with your explicit equations for $u$ and $v$ in the coordinate system $(x,y)$. These are not the differential equations $(u,v)$ must satisfy in order for $J_{\delta,\beta}$ be integrable. The correct equations when $n>1$ (as I show in my answer) are $2n$ in number, but you have written down $3n$ equations, which is too many. Also, it is clearly not right for $n=1$ because, when $n=1$ all $J_{\delta,\beta}$ are integrable complex structures. You need to fix this, because it misleads readers. | |
| Feb 15, 2016 at 15:49 | comment | added | Robert Bryant | @Solar: Answers to your questions: (1) Yes. (2) I have entered the proof as Remark 2 below. (3) Actually, for the hypersurface $Z{\cdot}Z+1 = 0$, the functions $\delta$ and $\beta$ are functions of the type you mention (i.e., they are functions of $|v|^2$). However, for the generic hyperquadric that I wrote down, they are not. This actually follows from Remark 1, since the form for $E(x,y)$ that you wrote down is not invariant under the action of the affine group of $\mathbb{R}^{n+1}$, but you can just take an explicit example when $n=1$ and check this directly. | |
| S Feb 15, 2016 at 15:34 | history | suggested | Amir Baghban | CC BY-SA 3.0 |
Putting braces in the item 1 of the last rows, editing some typos.
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| Feb 15, 2016 at 15:22 | review | Suggested edits | |||
| S Feb 15, 2016 at 15:34 | |||||
| Feb 15, 2016 at 15:06 | history | edited | Todd Trimble | CC BY-SA 3.0 |
fixed up a mess
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| S Feb 15, 2016 at 14:41 | history | suggested | Amir Baghban | CC BY-SA 3.0 |
Regarding the answer I have the stated questions which I can't ask in my post because apparently I don't have enough points to post comments in the site.
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| Feb 15, 2016 at 13:50 | review | Suggested edits | |||
| S Feb 15, 2016 at 14:41 | |||||
| Feb 13, 2016 at 0:17 | comment | added | Robert Bryant | @Solar: I noticed that you have not responded to the answer that I posted. Do you need more details, have doubts or objections, etc.? Let me know and I will do my best to answer them. | |
| Feb 11, 2016 at 10:59 | history | edited | user9072 | CC BY-SA 3.0 |
changed title, further improvements might make sense
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| Feb 9, 2016 at 14:54 | answer | added | Robert Bryant | timeline score: 11 | |
| S Feb 9, 2016 at 6:30 | history | suggested | Amir Baghban | CC BY-SA 3.0 |
I am the original poster and I added the necessary information to the questions
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| Feb 9, 2016 at 6:03 | review | Suggested edits | |||
| S Feb 9, 2016 at 6:30 | |||||
| Feb 9, 2016 at 0:07 | history | edited | Robert Bryant |
edited tags
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| Feb 8, 2016 at 17:01 | review | First posts | |||
| Feb 8, 2016 at 17:39 | |||||
| Feb 8, 2016 at 17:00 | history | asked | Amir Baghban | CC BY-SA 3.0 |