Timeline for Surfaces all of whose geodesics are both closed and simple
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 20, 2013 at 14:21 | comment | added | Olga | The manifold should be smooth, otherwise there is an example in Besse book with closed geodesics, all of the same length except one ("an equator"). | |
| Mar 3, 2012 at 2:57 | comment | added | Anton Petrunin | @BS: you should assume something about the space, otherwise the space forms would give a counterexample. | |
| Feb 29, 2012 at 18:34 | comment | added | BS. | According to Berger's "Panoramic view of Riemannian geometry", p. 439, Gromoll and Grove proved in 1981 that the hypothesis that all geodesics are closed implies that they have the same length and are simple. | |
| Jun 19, 2010 at 1:45 | vote | accept | Joseph O'Rourke | ||
| Jun 18, 2010 at 14:07 | comment | added | Joseph O'Rourke | @Benoit: Thanks! I wonder from the title of that book (which I just ordered through Interlibrary Loan) if this class of manifolds is of interest: every closed geodesic is simple (but not necessarily every geodesic is closed). | |
| Jun 18, 2010 at 13:18 | comment | added | Benoît Kloeckner | Yes, many metrics have this property. There is a very detailled reference by Arthur Besse: Manifolds all of whose Geodesic are closed. | |
| Jun 18, 2010 at 13:17 | comment | added | Joseph O'Rourke | Thanks so much! V. Guillemin, The Radon transform on Zoll surfaces. Advan. Math. 22 (1976), pp. 85–119. I will have to retrieve that paper! | |
| Jun 18, 2010 at 13:09 | history | answered | Anton Petrunin | CC BY-SA 2.5 |