Abstract:
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.
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