Timeline for Probing a manifold with geodesics
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| S Jul 30, 2017 at 5:41 | history | suggested | Martin Sleziak |
removed deprecated (geometry) tag - see the tag info: http://mathoverflow.net/tags/geometry/info; if there are some other geometry-related tags which are suitable, please use some of them instead
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| Jul 30, 2017 at 5:08 | review | Suggested edits | |||
| S Jul 30, 2017 at 5:41 | |||||
| Jul 29, 2017 at 15:13 | comment | added | Michael | @HansStricker: if $M$ was simply connected the 1st loop would be retractible and therefore removal of it would separate $M$ into two pieces. The 2nd circle would travel from one piece to another at the transverse intersection, and in order to come back would have to intersect the 1st loop once more. | |
| Jul 29, 2017 at 13:11 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image link broken; now fixed.
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 12, 2013 at 9:34 | comment | added | Hans-Peter Stricker | @fedja: Is it, that one just has to show that two such loops - with just one intersection which is not a "kissing" - cannot be homotopic? Such that the fundamental group is not trivial? | |
| Mar 10, 2013 at 16:34 | comment | added | Hans-Peter Stricker | Is there someting new concerning fedja's interesting comment, in the meanwhile? | |
| Dec 17, 2011 at 17:28 | answer | added | Robert Bryant | timeline score: 22 | |
| Nov 29, 2011 at 16:04 | answer | added | Richard Montgomery | timeline score: 9 | |
| Nov 23, 2011 at 11:52 | comment | added | Dror Atariah | By recording the geodesics intersections you can collect information on the cut locus of the surface, which in turn provide you with insight into the topology of the surface. | |
| Nov 22, 2011 at 21:39 | comment | added | Joseph O'Rourke | @fedja: Excellent point (thanks!), but I am uncertain what conclusions could be drawn... | |
| Nov 22, 2011 at 21:35 | comment | added | Joseph O'Rourke | @Noam: Perhaps you mean, "other than homeomorphic to a sphere," rather than precisely a sphere. Because Zoll's surface has the property that every geodesic is closed and simple (mathoverflow.net/questions/28622). I am surprised to learn that typical geodesics are dense. I thought there would generally be "unreachable" sections. Glad to have my faulty intuition corrected! | |
| Nov 22, 2011 at 18:17 | comment | added | fedja | Having two orthogonal circles of different radii (like on the torus) already tells you something. You do not even need geodesics; if two closed curves on the manifold have just one intersection and it is transversal, that certainly should ring some bells though I'd better leave it to the many topologists here to tell what and how exactly can be derived from it. :). | |
| Nov 22, 2011 at 17:35 | comment | added | Noam D. Elkies | I thought that on a compact $M$, other than a sphere, the typical geodesic is dense in $M$. This would imply that if you really have the image of all of $[0,\infty)$ under a geodesic path then you know $M$ exactly and can thus deduce its genus and any other property you care about. | |
| Nov 22, 2011 at 16:07 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |