Timeline for Is every closed curve in 3D a geodesic on a genus-0 surface?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| May 25, 2015 at 11:48 | vote | accept | Joseph O'Rourke | ||
| May 23, 2015 at 21:55 | comment | added | Andy Putman | This "answer" (by John Robertson, now converted to a comment by a moderator) doesn't really have any mathematical content, and in any case is wrong. Every unknotted simple closed curve in $\mathbb{R}^3$ lies on the surface of a smooth ball, but my answer identifies a series of obstructions that prevent you from making it a geodesic. | |
| May 23, 2015 at 20:21 | comment | added | The Masked Avenger | Thus the comment above on balloon and pinch. If genus 1 were allowed, there would be almost nothing to work on. | |
| May 23, 2015 at 20:01 | comment | added | John Robertson | No, this is no proof. Yes, it is probably the way you could make a proof. As long as you can embed the curve in the surface of a smooth ball the answer would be yes by "inflating" the sphere a small amount except for a "trench" which contains the curve. The base of the trench also needs to be modified by being angled to be perpendicular to the plane of a best fit circle to the curve at that point. No time for anything better. But I think that is the outline of how you prove it. | |
| May 23, 2015 at 5:29 | answer | added | Andy Putman | timeline score: 24 | |
| May 23, 2015 at 2:29 | comment | added | The Masked Avenger | Take a deflated balloon, put it in the middle, and inflate it. If you do it carefully, the balloon will be pinched by the curve for some of the curve. Elongate the balloon to maintain the pinch. | |
| May 23, 2015 at 2:14 | comment | added | Andy Putman | I edited the question to correct the spelling of my name in the revision (sigh). | |
| May 23, 2015 at 2:14 | history | edited | Andy Putman | CC BY-SA 3.0 |
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| May 23, 2015 at 1:37 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
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| May 23, 2015 at 1:11 | comment | added | Joseph O'Rourke | @AndyPutman: Thanks for the quick answer and cite of Schoenflies. Revised now to focus on unknotted curves. | |
| May 23, 2015 at 1:10 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
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| May 23, 2015 at 1:04 | answer | added | Igor Rivin | timeline score: 8 | |
| May 23, 2015 at 1:03 | comment | added | Andy Putman | It cannot be knotted: the Schoenflies theorem says that such a $\gamma$ bounds a disc in $S$, and is thus the unknot. | |
| May 23, 2015 at 0:47 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |