The number of exceptional orbits of a pseudofree circle action on S 5

Abstract

Let M be a closed Riemannian manifold of dimension 5 which admits a Riemannian metric of nonnegative sectional curvature. The aim of this short paper is to show that under certain lower bound of the orders of isotropy subgroups, every pseudofree and isometric S1-action on M cannot have more than five exceptional circle orbits. As a consequence, we conclude that a pseudofree and isometric S1-action on a 5-sphere S5 with a Riemannian metric of nonnegative sectional curvature cannot have more than five exceptional circle orbits. This gives a result related to the Montgomery–Yang problem. In addition, we also give some further related result about nonnegatively curved manifolds of dimension 5 with an isometric but not necessarily pseudofree circle action.