Timeline for Are there Ricci-flat riemannian manifolds with generic holonomy?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 4, 2010 at 0:04 | vote | accept | José Figueroa-O'Farrill | ||
| Mar 3, 2010 at 2:32 | comment | added | Igor Belegradek | Jose, do not worry about these things; my contribution was tiny. | |
| Mar 2, 2010 at 23:53 | comment | added | José Figueroa-O'Farrill | I'm minded to accept this answer, but I will wait a little while to see if anyone else has anything to say whether some further progress has been made on this question. | |
| Mar 2, 2010 at 17:15 | comment | added | José Figueroa-O'Farrill | @Igor: I've just taken a look at Petersen. Thanks. So indeed this is the "euclideanisation" of the lorentzian Schwarzschild solution. I will be updating the question to incorporate your answer. Many thanks. | |
| Mar 2, 2010 at 17:01 | comment | added | Igor Belegradek | Jose, the Schwarzschild metric I know is Riemannian. :) Look e.g. in Petersen's "Riemannian Geometry" text for detailed study of Riemannian Schwarzschild metric | |
| Mar 2, 2010 at 16:56 | comment | added | Igor Belegradek | Jose, yes this is the book. Another standard source is "Surveys in Differential Geometry: Essays on Einstein manifolds" by International Press which was also published a decade ago. I do not think there have been any major breakthroughs since then in regard to your question. | |
| Mar 2, 2010 at 16:55 | comment | added | José Figueroa-O'Farrill | Another question: the Schwarzschild metric I know is lorentzian, so it has holonomy $\mathrm{SO}(3,1)$. Hence this answers the bonus question! This prompts another question: are there noncompact Ricci-flat riemannian manifolds with generic holonomy? | |
| Mar 2, 2010 at 16:48 | comment | added | José Figueroa-O'Farrill | Is the book by Berger that you mention the same as A panoramic view of Riemannian geometry? | |
| Mar 2, 2010 at 15:28 | comment | added | Deane Yang | Berger's book is pretty old by now, but I think the situation remains as you describe. As far as I know, every construction of a closed Ricci-flat manifold proceeds by building a metric with a holonomy group that implies Ricci-flat. The holonomy assumption implies algebraic conditions on the curvature tensor that in turn play crucial roles in the elliptic estimates used to prove existence. In some sense, you're replacing a second order elliptic PDE by a stronger first order system that implies the second order PDE. | |
| Mar 2, 2010 at 15:22 | history | answered | Igor Belegradek | CC BY-SA 2.5 |