As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on the diagonal by $U(1) \times U(1)$, embedded in the diagonal as $(e^{i \theta_1}, e^{i \theta_2}, e^{-i(\theta_1 + \theta_2})$ then we get the full flag manifold of $SU(3)$. However, we can also embed $U(1)$ into $SU(3)$ on the diagonal as $(e^{i \theta}, e^{-i \theta}, 1)$. What is the corresponding quotient? Is it somehow pathological?
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The various different ways that $\mathrm{U}(1)\simeq S^1$ can appear as a subgroup of $\mathrm{SU}(3)$ are indexed by a lattice of rank $2$, and the $7$-dimensional quotients are now known as Aloff-Wallach manifolds, after a paper by Simon Aloff and Nolan Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures (Bulletin of the AMS 81 (1975), 1–222). Your particular one is not singular in any way, nor do I think it has any particular pathologies.
In recent years, a great deal has been learned about Aloff-Wallach manifolds, more than I can recount here. I suggest a literature search.
answered Apr 18, 2014 at 16:22
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2$\begingroup$ 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. $\endgroup$ Commented Apr 18, 2014 at 16:28
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1$\begingroup$ @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. $\endgroup$ Commented Apr 18, 2014 at 16:39
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$\begingroup$ 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. $\endgroup$ Commented Apr 30, 2014 at 22:58