The sectional curvature remains positive when taking quotients by certain nonfree actions

TL;DR

This paper investigates conditions under which positive sectional curvature is preserved when taking quotients by certain nonfree isometric actions of Lie groups, specifically for actions of S^1 and S^3.

Contribution

It demonstrates that quotient spaces retain positive sectional curvature under specific conditions involving nonfree group actions, extending previous understanding of curvature preservation.

Findings

Quotients by S^1 and S^3 actions can have positive sectional curvature.

Positive curvature is preserved if the original metric is positive on planes orthogonal to orbits.

The paper constructs smooth quotient spaces with positive sectional curvature.

Abstract

We study some cases when 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 $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space carries 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.