Hopf fibrations are characterized by being fiberwise homogeneous

TL;DR

This paper characterizes Hopf fibrations of spheres by their fiberwise homogeneity, showing that this property uniquely identifies them among sphere fibrations, and extends the characterization to local fiberwise homogeneity in the 3-sphere case.

Contribution

The paper proves that Hopf fibrations are uniquely characterized by fiberwise homogeneity, both globally and locally, among all sphere fibrations by smooth subspheres.

Findings

Hopf fibrations are characterized by fiberwise homogeneity.

Fiberwise homogeneity uniquely identifies Hopf fibrations among sphere fibrations.

Local fiberwise homogeneity implies a fibration is part of a Hopf fibration in the 3-sphere case.

Abstract

Heinz Hopf's famous fibrations of the 2n+1-sphere by great circles, the 4n+3-sphere by great 3-spheres, and the 15-sphere by great 7-spheres have a number of interesting properties. Besides providing the first examples of homotopically nontrivial maps from one sphere to another sphere of lower dimension, they all share two striking features: (1) Their fibers are parallel, in the sense that any two fibers are a constant distance apart, and (2) The fibrations are highly symmetric. For example, there is a fiber-preserving isometry of each total space which takes any given fiber to any other one. Hopf fibrations have been characterized up to isometry by the first property above, initially among all fibrations of spheres by great subspheres, and later in the stronger sense among all fibrations of spheres by smooth subspheres. In this paper, we show that the Hopf fibrations are also characterized by their "fiberwise homogeneity" expressed above in (2), and in the strong sense among all fibrations of spheres by smooth subspheres. In the special case of the 3-sphere fibered by great circles, we prove something stronger. We prove that a fibration of a connected open set by great circles which is locally fiberwise homogeneous is part of a Hopf fibration.