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?



Quotienting $SU(3)$ by $U(1)$?
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.


