Timeline for Question on $\alpha-$Einstein manifolds
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 2, 2017 at 5:16 | vote | accept | C.F.G | ||
| Sep 1, 2017 at 17:49 | vote | accept | C.F.G | ||
| Sep 2, 2017 at 5:16 | |||||
| Aug 15, 2017 at 13:02 | vote | accept | C.F.G | ||
| Aug 15, 2017 at 13:03 | |||||
| Aug 14, 2017 at 11:37 | comment | added | Otis Chodosh | You might consider the "harmonic map - Ricci flow" coupled system (see arxiv.org/pdf/0912.2907.pdf). If the target is $\mathbb{R}$, the fixed points will locally be the same as your "$\alpha$-Einstein" metrics, with the added condition that $\alpha$ is harmonic. Depending on your motivation, adding this assumption might be natural. There is a very important connection with general relativity as well, as discussed in e.g. intlpress.com/site/pub/files/_fulltext/journals/cag/2008/0016/… . | |
| Aug 14, 2017 at 10:49 | answer | added | C.F.G | timeline score: 1 | |
| Aug 1, 2017 at 13:00 | comment | added | C.F.G | You are right. I want any possible answer. Thanks for your comment. | |
| Jul 31, 2017 at 14:07 | comment | added | Robert Bryant | Note that, in the case of dimension $3$, this notion of '$\alpha$-Einstein' is simply the condition that the Ricci tensor have a double eigenvalue (namely, the function $a$). There's not much to say about such metrics in general. They aren't real-analytic in harmonic coordinates in general; for example, if a Riemannian $3$-manifold has full rotational symmetry about some point, then its Ricci tensor will have at least a double eigenvalue everywhere. You could probably write down a flow that has such metrics as fixed points, but it won't have any good properties. | |
| Jul 31, 2017 at 9:43 | history | edited | C.F.G | CC BY-SA 3.0 |
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| Jul 31, 2017 at 9:25 | history | edited | C.F.G |
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| Jul 31, 2017 at 9:00 | history | asked | C.F.G | CC BY-SA 3.0 |