Timeline for Curves of constant curvature on an ellipsoid
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 3, 2021 at 9:48 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Removed a speculative statement that I no longer think is likely to be true.
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| Feb 28, 2021 at 14:33 | comment | added | Vishesh | @RobertBryant. I am sorry, Prof Bryant. I misunderstood the statement as claiming the converse as you rightly guessed. Thank you for clarifying. | |
| Feb 27, 2021 at 11:35 | comment | added | Robert Bryant | @Vishesh: I don't understand your question. Darboux' claim is that if all of the curves of constant (nonzero) geodesic curvature are closed then the surface has constant Gauss curvature. He does not claim the converse, that all the curves of constant (non-zero) geodesic curvature on a surface of constant curvature are closed, and, of course, this is not true. For example, if one takes the simply connected cover of the flat plane or the hyperbolic disk minus a point or the sphere minus two antipodal points, then the circles around the missing point will lift to non-closed curves in the cover. | |
| Feb 27, 2021 at 5:52 | comment | added | Vishesh | @RobertBryant This question is 3 years old, but I had to ask as I am confused about Darboux's claim. If one takes the tangential developable of a circular helix, then the surface has helices all of which are curves of constant geodesic curvature on that surface. But they don't close up. What am I missing here?? | |
| Jul 25, 2017 at 8:20 | comment | added | Robert Bryant | @IanAgol: You are right that there is more to say, but I think it can be completed as follows: Take the tautological forms $(\omega_1,\omega_2,\omega_{12})$ on the unit circle bundle $B$ of the surface, and on $B\times\mathbb{R}$, consider the integral curves of the system $dt=\omega_2=\omega_1-t\omega_{12}=0$. By hypothesis, these are all closed, with the $t=0$ curves being the unit circles themselves. You can now compute the derivative of the Poincaré return map around the $t=0$ curves and find that it is the identity. Thus, the curves close up as simple curves on the surface for $t$ small. | |
| Jul 24, 2017 at 18:04 | comment | added | Ian Agol | I don't quite get the last deduction of Darboux' claim from Rick's paper. Maybe all constant curvature curves close up, but they might spin around many times before closing up (so are immersed, rather than embedded). I didn't look at Darboux' book, so maybe he meant simple closed curves. Of course, it seems implausible, but I didn't get the logic. | |
| Jul 24, 2017 at 13:16 | comment | added | Joseph O'Rourke | "Foliation by constant mean curvature spheres": PDF download. | |
| Jul 24, 2017 at 13:10 | history | answered | Robert Bryant | CC BY-SA 3.0 |