Timeline for Closed Semi-Riemannian manifolds with non-compact isometry group
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 25, 2018 at 14:37 | vote | accept | JS. | ||
| Aug 24, 2018 at 11:39 | answer | added | Uri Bader | timeline score: 4 | |
| Aug 24, 2018 at 11:06 | comment | added | Uri Bader | regarding left and right: I believe it is absolutly symmetric and you have to break this symmetry by an arbitrary choice - as usual. Regarding PSL vs. SL: again it is just a matter of convention. Maybe it is more elegant to use PSL_2 as the base space because it is adjoint. I do not think it matters much. | |
| Aug 23, 2018 at 16:30 | comment | added | JS. | I have a (possibly stupid) question about the paper. I wonder if there is a reason why the author uses quotients $\Gamma \backslash \widetilde{PSL}(2,\mathbb{R})$ instead of $\widetilde{SL}(2,\mathbb{R}) / \Gamma$?. The first thing I ask myself is, if both constructions yield isometric spaces (so does it matter if we take the left or right quotient)? And the second question is: is there any reason he writes $\widetilde{PSL}$ instead of $\widetilde{SL}$? As far as I know those to spaces are the same (since $SL$ covers $PSL$ two sheeted), but during the whole paper he writes $\widetilde{PSL}$. | |
| Aug 15, 2018 at 12:00 | comment | added | Uri Bader | No, not all signatures appear by the construction above. For example the lorentzian signature $(n,1)$ appears only for $n=2$. In fact the only non-compact simple group acting on a compact lorentzian manifold is $SL_2(\mathbb{R})$ (up to local isomorphism). | |
| Aug 15, 2018 at 11:56 | comment | added | Uri Bader | take a look here: arxiv.org/pdf/1804.08695.pdf | |
| Aug 15, 2018 at 8:36 | comment | added | JS. | Thank you. I will take a look at it. I have a question to your construction: Can we say anything about the signature of the PR-metric we obtain from the Killing form? In the case of $SL(2,\mathbb{R})$ it has signature $(2,1)$. Do all signatures appear? | |
| Aug 14, 2018 at 19:46 | comment | added | Uri Bader | take a look here: ihes.fr/~/gromov/PDF/1[74].pdf | |
| Aug 14, 2018 at 18:19 | history | edited | JS. | CC BY-SA 4.0 |
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| Aug 14, 2018 at 15:21 | comment | added | JS. | Thanks for your comment. I adjusted my question accordingly. To your construction: so whenever $G$ is non-compact this yields an example of a compact PS-manifold $G/\Gamma$ with non-compact isometry group (and if $G$ is compact itself, then the metric on $G$ would be Riemannian), right? | |
| Aug 14, 2018 at 15:11 | history | edited | JS. | CC BY-SA 4.0 |
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| Aug 14, 2018 at 14:08 | comment | added | Uri Bader | There are quite a few results about pseudo-Riemannian manifolds. Could you please be more specific about what is it that you want to get? In any case, here is a nice construction of a PR-manifold with a reach group of isometries. Let $G$ be a semisimple group and use left translations of the killing form on $T_eG$ to obtain a PR-structure on $G$. By the conjugation invariance of the form, this structure is also right invariant. Take a cocompact lattice $\Gamma<G$ and form the quotient structure on $G/\Gamma$. The left action of $G$ on this space will be by isometries. | |
| Aug 14, 2018 at 13:10 | history | asked | JS. | CC BY-SA 4.0 |