Highly curved orbit spaces (Q2414136)

The orbit space $X$ of an isometric Lie group action of cohomogeneity at least two on the unit sphere $S^n$, $n\geq 2$, is an Alexandrov space of curvature at least one. Denote by $\kappa_X$ the infimum of the sectional curvatures on the regular part of $X$. \par In previous work the author and \textit{A. Lytchak} [Geom. Dedicata 190, 135--142 (2017; Zbl 1380.57033)] showed that $\kappa_X$ is at most $4$. Moreover, it follows from that paper that $\kappa_X=1$ for ``most'' actions. \par In the paper under review it is made more precise what ``most'' means: It is shown that $\kappa_X>1$ if and only if the action is of one of five types.