Timeline for A riemannian manifold with finitely many closed contractible geodesics
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| May 12, 2013 at 8:17 | history | edited | user5810 | CC BY-SA 3.0 |
changed "three closed" to "three simple closed"
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| Oct 27, 2012 at 18:35 | comment | added | Igor Rivin | Yes, correct (re simple vs not)... | |
| Oct 27, 2012 at 14:18 | comment | added | Malte | I figured out the problem here. The result in Igor Rivin's post is (almost) the Ljusternik-Schnirelmann theorem, which asserts that an ellipsoid with axes very close to the round sphere has exactly three simple closed gedesics (namely, the intersections of the ellipsoid and the coordinate planes). Moreover, it seems that Calabi once conjectured that there are no compact mf. with finitely many closed geodesics. (This does not mean, however, that there are no non-simply-connected mf. with only finitely many contractible closed geodesics). I guess the question has no answer (yet). | |
| Oct 26, 2012 at 20:51 | comment | added | Malte | This looks/sounds convincing. However, if this were true, it would contradict the theorem I mentioned above. The theorem is due to Franks-Bangert and asserts that every riemannian metric on $S^2$ has infinitely many prime closed geodesics. The "prime" relation is the same as the geometric distinction: $c_0$ is prime if it is not a multiple of another closed geodesic. | |
| Oct 26, 2012 at 19:57 | vote | accept | Malte | ||
| Oct 26, 2012 at 19:57 | |||||
| Oct 26, 2012 at 19:56 | vote | accept | Malte | ||
| Oct 26, 2012 at 19:57 | |||||
| Oct 26, 2012 at 19:56 | vote | accept | Malte | ||
| Oct 26, 2012 at 19:56 | |||||
| Oct 26, 2012 at 16:38 | history | answered | Igor Rivin | CC BY-SA 3.0 |