The Sectional Curvature Remains Positive When Taking Quotients by Certain Nonfree Actions

We study some cases in which the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ for which the quotient space can be endowed with a smooth structure by means of the fibrations $S^3/S^1\simeq S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space possesses a metric of positive sectional curvature provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.