Timeline for Geodesic transformations of the complex projective plane
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 2, 2013 at 11:21 | comment | added | Ben McKay | The same proof then proves that any homeomorphism of the octave projective plane or quaternionic projective plane which preserves geodesics is an isometry. It is nice how the proof breaks for the real projective plane. | |
| Aug 2, 2013 at 11:15 | comment | added | Ben McKay | I keep referring to earlier work that proves smoothness of homeomorphisms preserving projective lines. The proof: Bödi, Richard; Kramer, Linus On homomorphisms between generalized polygons. Geom. Dedicata 58 (1995), no. 1, 1–14. | |
| Aug 2, 2013 at 11:04 | comment | added | Anton Petrunin | @Ben, Thank you --- that is exactly what I mean (and by the way my statement is correct; one only has to read it carefully). | |
| Aug 2, 2013 at 10:42 | comment | added | Ben McKay | If you take any two points of the complex projective plane, there is either a unique geodesic connecting them, or else the union of the geodesics is a projective line. Therefore geodesic preserving homeomorphisms preserve projective lines, and therefore are smooth projective transformations (from my work, or earlier work on smooth projective planes). Then Anton's argument finishes the proof: all geodesic preserving homeos of the complex projective plane are isometries. | |
| Aug 2, 2013 at 8:26 | comment | added | alvarezpaiva | @Vladmir: that's not true: there are (at least ?) the Lagrangian real projective planes as well. | |
| Aug 2, 2013 at 7:48 | comment | added | Vladimir S Matveev | This is a comment on the comment of Mischa about "and the other way around is clearly faulse". The small correction of Anton that makes the argument correct is ``every totally geodesic two-dimensional submanifold of CP(n) is a projective line''. | |
| Aug 2, 2013 at 6:25 | comment | added | alvarezpaiva | @Misha: we're just at the point where we wait for Robert to give us the answer ... ;-) | |
| Aug 2, 2013 at 5:58 | comment | added | Misha | @alvarezpaiva: Yes, this is what my comment is about. | |
| Aug 2, 2013 at 5:56 | comment | added | alvarezpaiva | Hmm, I think I was a bit quick in my previous comment. I'm taking for granted that the only totally geodesic two-dimensional submanifolds in the complex projective plane are the complex lines and the Lagrangian real projective planes, but I do not really know if this is true. | |
| Aug 2, 2013 at 5:08 | vote | accept | alvarezpaiva | ||
| Aug 2, 2013 at 6:09 | |||||
| Aug 2, 2013 at 5:07 | comment | added | alvarezpaiva | @Anton: I thought about this line of attack, but I got stuck in proving that complex lines get mapped to complex lines. Oops, never mind: they are the only totally geodesic submanifolds in their homology class. They have to be mapped to each other. Thanks !! | |
| Aug 2, 2013 at 4:51 | comment | added | Misha | Anton: You need more work for the 1st part of the proof, since "and the other way around" is clearly false. | |
| Aug 2, 2013 at 3:01 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
deleted 4 characters in body
|
| Aug 2, 2013 at 2:55 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
added 244 characters in body
|
| Aug 2, 2013 at 2:13 | history | answered | Anton Petrunin | CC BY-SA 3.0 |