Timeline for Quotienting $SU(3)$ by $U(1)$?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2014 at 22:58 | comment | added | Renato G. Bettiol | For the sake of reference, the Aloff-Wallach spaces $SU(3)/S^1_{k_0,k_1,k_2}$ are a special type of Eschenburg space, and many papers only refer to the latter more general class when computing topological invariants, etc. So perhaps it is also worth searching for "Eschenburg space" if you don't find what you want. A general Eschenburg space is a biquotient $SU(3)//S^1_{k,l}$, where $S^1_{k,l}\subset SU(3)\times SU(3)$ is a circle with slopes $(k_0,k_1,k_2)$ and $(l_0,l_1,l_2)$ inside the product of the maximal tori. | |
| Apr 18, 2014 at 16:42 | vote | accept | Dontok Bartalez | ||
| Apr 18, 2014 at 16:39 | comment | added | Robert Bryant | @FrancescoPolizzi: I wouldn't say, though, that that qualifies these spaces as 'singular' or 'pathological'. Indeed, it appears that the most pathological thing about the OP's particular example is that the methods of Aloff and Wallach do not suffice to produce a metric of positive curvature on that particular A-W manifold, while they do for all of the other A-W manifolds. | |
| Apr 18, 2014 at 16:28 | comment | added | Francesco Polizzi | In arxiv.org/abs/hep-th/0108245, page 16, the authors claim that the Aloff-Wallach manifold of the question (which is $N(1, -1)$ in their notation) is essentially the only one with a unique Einstein metric, whereas all the others carry two inequivalent Einstein metrics. So it can be considered special in this sense. | |
| Apr 18, 2014 at 16:22 | history | answered | Robert Bryant | CC BY-SA 3.0 |