Timeline for Geodesic transformations of the complex projective plane
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 2, 2013 at 10:33 | vote | accept | alvarezpaiva | ||
| Aug 2, 2013 at 10:31 | comment | added | alvarezpaiva | Thanks for you patience. I think I'm really thick today. @Ben: maybe it has to do with me being really thick today, but I can't see a clear-cut, easy argument characterizing the complex lines among the totally geodesic submanifolds of $\mathbb{CP}^n$. See my comments to Anton's answer. | |
| Aug 2, 2013 at 10:19 | comment | added | Vladimir S Matveev | Juan-Carlos, projective Lichnerowicz-Obata conjecture says that on a closed Riemannian manifold of nonconstant sectional curvature every infintesimal projective symmetry is a Killing vector field. It therefore can not be applied to the standard sphere or to the standard real projective space because they do have constant sectional curvature | |
| Aug 2, 2013 at 9:50 | comment | added | Ben McKay | I once wrote a paper called Smooth projective planes, which proved that the continuous maps which preserve orientation and take lines to lines are diffeomorphisms. It turned out that this was already known in the literature of topological projective planes. Using the fact that the geodesics of the complex projective plane lie on the projective lines, you can easily show that the homeomorphisms preserving orientation and geodesics are complex projective transformations. But then preserving geodesics is actually stronger, so they must be isometries as indicated below. | |
| Aug 2, 2013 at 8:30 | comment | added | alvarezpaiva | @Vladimir: Thanks !! I get the argument now. However, one can also make this argument for the real projective space and yet there are lots of projective transformations that are not isometries. | |
| Aug 2, 2013 at 8:24 | vote | accept | alvarezpaiva | ||
| Aug 2, 2013 at 8:26 | |||||
| Aug 2, 2013 at 8:14 | comment | added | Vladimir S Matveev |
Juan-Carlos, there were two mainstreams in the classical (<1990) theory of projectively equivalent metrics: the french'' (+ Lie) and japanese'' studied mostly infinitesimal projective transformations and ``soviet'' mostly projectively equivalent metrics. In the case your metric has a big group of symmetries, the results of both groups can be used by the following simple observation that I also explained in my answer: if the metrics are projectively equivalent, then an isometry of the first is projective transformation of the second.
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| Aug 2, 2013 at 7:43 | answer | added | Vladimir S Matveev | timeline score: 5 | |
| Aug 2, 2013 at 7:39 | comment | added | alvarezpaiva | @Misha: See section 1.3 in Vladimir"s paper. | |
| Aug 2, 2013 at 6:24 | comment | added | Misha | True, but, still, might be worth checking. | |
| Aug 2, 2013 at 6:18 | comment | added | alvarezpaiva | Misha: thanks for the reference. However, most and probably all (?) of this classic work relies on the existence of "infinitesimal" projective transformations. Here I would like to know if there is just one non-isometric transformation that maps geodesics to geodesics. | |
| Aug 2, 2013 at 5:57 | comment | added | Misha | There is a vast literature on "projective maps", going back to Sophus Lie, see e.g. www.minet.uni-jena.de/~matveev/Datei/lichnerowicz.ps and references there. | |
| Aug 2, 2013 at 5:42 | history | edited | alvarezpaiva | CC BY-SA 3.0 |
added 34 characters in body
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| Aug 2, 2013 at 5:08 | vote | accept | alvarezpaiva | ||
| Aug 2, 2013 at 6:09 | |||||
| Aug 2, 2013 at 2:13 | answer | added | Anton Petrunin | timeline score: 4 | |
| Aug 1, 2013 at 22:11 | history | asked | alvarezpaiva | CC BY-SA 3.0 |