Curvature and symmetry of Milnor spheres

TL;DR

This paper investigates the geometry of cohomogeneity one manifolds, demonstrating that certain group actions extend to these manifolds, leading to new non-negative curvature metrics on spheres and exotic spheres, and revealing infinitely many almost free actions on S^7.

Contribution

It proves that principal SO(3) and SO(4) actions extend to cohomogeneity one actions on bundles over S^4, enabling construction of many non-negatively curved metrics and actions on exotic spheres.

Findings

Every vector and sphere bundle over S^4 admits a complete non-negative curvature metric.

Existence of infinitely many non-negatively curved metrics on 7-dimensional exotic spheres.

Infinite almost free SO(3) actions on S^7 preserving the Hopf fibration.

Abstract

In this paper we explore the geometry and topology of cohomogeneity one manifolds, i.e. manifolds with a group action whose principal orbits are hypersurfaces. We show that the principal group action of every principal SO(3) and SO(4) bundle over S^4 extends to a cohomogeneity one action. As a consequence we prove that every vector bundle and every sphere bundle over S^4 has a complete metric with non-negative curvature. It is well known that 15 of the 27 exotic spheres in dimension 7 can be written as S^3 bundles over S^4 in infinitely many ways, and hence we obtain infinitely many non-negatively curved metrics on these exotic spheres. A further consequence will be that there are infinitely many almost free actions by SO(3) on S^7, i.e. all isotropy groups are finite. These actions preserve the Hopf fibration S^3 -> S^7 -> S^4 but do not extend to the disc D^8. We also construct infinitely many such actions on the 15 exotic 7-spheres mentioned above.