Hopf fibrations are characterized by being fiberwise homogeneous

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.