Timeline for Is it true that the geodesics on SO(n) and SU(n) are closed?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2013 at 17:02 | comment | added | Misha | Reza: If all geodesics are periodic and manifold M is simply connected then homology ring of M is generated by a single element. This will help you to rule out many examples you might be thinking about (like flag manifolds except for the ones which are CROSS). | |
| Mar 31, 2013 at 14:17 | comment | added | alvarezpaiva | Reza, if you're interested in manifolds all of whose geodesics are closed, there is a book by Arthur Besse that has everything that was known about the topic up the the 80's. I don't recall if there is a result saying the the only homogeneous examples are the standard ones, but I think that must be the case. Just don't try to contact the author, he doesn't respond well to email. | |
| Mar 31, 2013 at 6:05 | comment | added | Reza Rezazadegan | Thank you guys. I got the answer to my original question. However what if we mod out by the maximal torus? For example $U(n)/T^n$ is a flag variety and as the answe to to this Mathoverflow question mathoverflow.net/questions/7750/geodesics-on-a-grassmannian explains, the geodesics on it are given by $exp(tX)$ where $X$ lies in a complement of the Lie algebra of the maximal torus. | |
| Mar 30, 2013 at 16:06 | comment | added | alvarezpaiva | Just spelling out Ben's comment: compact Lie groups are compact Riemannian symmetric spacesand as such contain totally geodesic flat tori. The maximal dimension of these tori is called the rank of the space. If the rank is greater than one, the flat tori will have geodesics that wind around without closing. On the other hand the list of rank-one symmetric spaces is very short: spheres, projective spaces over the real, the complex numbers, and the quaternions, as well as the Cayley plane. The only Lie groups in the list are $SU(2) = S^3$ and $SO(3)$ which is three-dimensional projective space. | |
| Mar 30, 2013 at 14:31 | comment | added | Ben Webster♦ | Reza- This won't work outside $SU(2)$. In a compact Lie group of rank $>1$, you can find irrational geodesics in the torus. | |
| Mar 30, 2013 at 14:25 | comment | added | Reza Rezazadegan | OK, thanks. But the round metric on $SU(2)\cong S^3$ has closed geodesics. | |
| Mar 30, 2013 at 13:21 | history | edited | John Jiang | CC BY-SA 3.0 |
deleted 52 characters in body
|
| Mar 30, 2013 at 13:14 | history | answered | John Jiang | CC BY-SA 3.0 |