Timeline for Are geodesics necessarily embedded?
Current License: CC BY-SA 4.0
8 events
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| Jan 12, 2023 at 9:56 | comment | added | Leo Moos | You're right - I overlooked that part. Maybe you can glue in a smarter cap that would make the meridians self-intersect in there. Seems plausible, but it's purely conjectural. | |
| Jan 12, 2023 at 9:24 | comment | added | Dmitrii Korshunov | @LeoMoos >>the rotationally symmetric example you can also make closed by capping it on both ends.<< but all the meridians of a surface of revolution are geodesics, so there are infinitely many closed simple geodesics at each step. | |
| Jan 12, 2023 at 9:13 | comment | added | Leo Moos | OK, thanks for the clarification! It seems like an interesting question to ask how many simple, closed geodesics there are say $C^1$ or $C^2$ generically. (This means that you can't glue in small 'mushroom-like' caps with short geodesic loops anymore.) The rotationally symmetric example you can also make closed by capping it on both ends. I included this because it can be made to converge smoothly, unlike examples with small caps. (In higher dimensions, manifolds generically contain infinitely many embedded minimal surfaces; the contrast would be interesting.) | |
| Jan 12, 2023 at 6:10 | comment | added | Dmitrii Korshunov | @LeoMoos also a compact counterexample to the upper-semicontinuity can be like this: take an ellipsoid with exactly three simple geodesics and modify it in a small region outside of the union of simple geodesics such that there is one more simple geodesic and shrink this modified part. | |
| Jan 12, 2023 at 5:04 | history | edited | Dmitrii Korshunov | CC BY-SA 4.0 |
added 7 characters in body
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| Jan 12, 2023 at 4:10 | comment | added | Dmitrii Korshunov | @LeoMoos thank you, you are right. the statement that an $C^0$ open dense set of metrics can't be as i claimed. nevertheless i wonder if one still can make sense of the quustion "what is average/generic number of simple geodesics" [refining topology, extending the notion of genericity etc.] | |
| Jan 11, 2023 at 8:54 | comment | added | Leo Moos | 'For a generic metric on $S^2$ there are only three such geodesics.' Is this accurate? Certainly there's at least three, but only three is a much stronger claim. (You can find arbitrarily $C^0$-small perturbations of any metric with many geodesics.) Also, is the number of simple closed geodesics really upper semicontinuous? What about something like a rotationally symmetric surface obtained by rotating $f =1+x^2(1+\epsilon \operatorname{sin}(1/\epsilon x))$ and letting $\epsilon \to 0$? | |
| Dec 5, 2022 at 17:54 | history | answered | Dmitrii Korshunov | CC BY-SA 4.0 |