Timeline for Geodesic transformations of the complex projective plane
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 2, 2013 at 10:33 | vote | accept | alvarezpaiva | ||
| Aug 2, 2013 at 9:54 | comment | added | Vladimir S Matveev | You would not love the initial proof which is bases on tensor calculations. It is quite short though and could be put on the 2-page Doklady paper which I do not have by hand and can not check therefore. Actually, the review in MathReviews already contain some information. If you want a conceptual proof, a symmetric space is einstein so the Thomas cone connection is actually the Levi-Civita connection of a certain metric on the cone over the manifold. Projectively related metrics correspond to parallel (0,2) tensors for this cone metric and because of many symmetries they do not exist | |
| Aug 2, 2013 at 8:39 | comment | added | alvarezpaiva | thanks again. I think I need to see this theorem of Sinjukov (is there a proof somewhere? Dokl. is just for announcements, right?). | |
| Aug 2, 2013 at 8:24 | vote | accept | alvarezpaiva | ||
| Aug 2, 2013 at 8:26 | |||||
| Aug 2, 2013 at 8:15 | comment | added | Vladimir S Matveev | No, I dont do it in my answer. The result of Sinjukov is local, works in all signatures, and is about projectively equivalent metrics and not about projective transformations or connected groups of projective transformations. The arguments in my asnwer using Lichnerowciz-Obata relies on the existence of a big group of isometries of the metrics in your question. If there exists a (noninfinitesimal) projective transformation, then the pullback of the killing vector fields are projective vector fields and one can use Licherowicz-Obata. The Kähler result is also about projective equivalence. | |
| Aug 2, 2013 at 8:08 | comment | added | alvarezpaiva | In your paper on the Lichnerowiz-Obata conjecture you explicitly warn that you need the hypothesis of a connected Lie group of projective transformations. Here I'm asking for just the existence of one non-trivial projective transformation that can be very far from the identity. In your comments do you assume that we have a connected Lie group of projective transformations? | |
| Aug 2, 2013 at 7:55 | comment | added | Vladimir S Matveev | I do not understand your comment. If you are asking whether in my answer I assumed that the projective transformation is actually an infinitesimal projective transformation, I did not do it and spoke about projectively equivalent metrics only. If you would like me to tell you more assuming the existence of an infinitesimal projective transformation, then globally (in the riemannian case) it is in arXiv:math/0407337 and locally in Solodovnikov 1956 in dimensions >2 and in arXiv:0802.2344 + arXiv:0705.3592 in dimension =2. In the case I misunderstood your comment, please explain | |
| Aug 2, 2013 at 7:46 | comment | added | alvarezpaiva | In the question there is no connected Lie group, no infitesimal projective transformations. Could you please specify how much of what you are saying carries through to this situation? | |
| Aug 2, 2013 at 7:43 | history | answered | Vladimir S Matveev | CC BY-SA 3.0 |