  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.
