Timeline for Kähler potentials that depend only on geodesic distance
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 12, 2012 at 0:36 | vote | accept | Oliver Jones | ||
| Jul 12, 2012 at 0:36 | answer | added | Oliver Jones | timeline score: 0 | |
| Jul 9, 2012 at 21:15 | history | edited | Robert Bryant |
added the differential geometry tag
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| Jul 9, 2012 at 12:27 | answer | added | Robert Bryant | timeline score: 6 | |
| Jul 4, 2012 at 0:02 | comment | added | Oliver Jones | @Misha: Correct Misha. The Fubini-Study, Begman, and the Euclidean metrics all have potentials of this form. @Yang: I'm speaking locally here; obviously the cut locus will need to be avoided. However, the base point $p$ is arbitrary; you simply avoid the cut locus of $p$. I've thought about this some more and I realized that the potential is also a function of $|z|^2=\sum_i|z_i|^2$. Here $(z_1,\cdots, z_n)$ are local coordinates. Rotationally symmetric metrics have this property. | |
| Jul 3, 2012 at 22:14 | comment | added | Deane Yang | Also, I am under the impression that the statement is not literally true for a compact manifold, since the Hessian of the distance function has to degenerate at the cut locus. It seems to me that the question needs to be stated more carefully. | |
| Jul 3, 2012 at 22:13 | comment | added | Deane Yang | For a hermitian symmetric space of constant curvature, this holds for any point $p$ (as defined in Misha's comment). It should not be difficult to give an example where it holds for only one particular choice of $p$ but not for others. So presumably the question is about manifolds where the potential can be written as a function of distance from a point $p$, for any choice of $p$? | |
| Jul 3, 2012 at 21:52 | comment | added | Misha | @Igor: He means that a potential for the metric is of the form $f(x)=h(d(p,x))$ where $p$ is a fixed point and $h$ is some function of one variable. | |
| Jul 3, 2012 at 18:47 | history | edited | user9072 | CC BY-SA 3.0 |
correction for umlaut
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| Jul 3, 2012 at 14:58 | comment | added | Igor Rivin | Could you elaborate on this question? How can a metric fail to be a function of geodesic distance? | |
| Jul 3, 2012 at 1:41 | history | asked | Oliver Jones | CC BY-SA 3.0 |