  In a 1983 paper with Frank Warner, we proved that the space of all great
circle fibrations of the 3-sphere S^3 deformation retracts to the subspace of
Hopf fibrations, and so has the homotopy type of a pair of disjoint
two-spheres. Since that time, no generalization of this result to higher
dimensions has been found, and so we narrow our sights here and show that in an
infinitesimal sense explained below, the space of all smooth oriented great
circle fibrations of the 2n+1 sphere S^(2n+1) deformation retracts to its
subspace of Hopf fibrations. The tools gathered to prove this also serve to
show that every germ of a smooth great circle fibration of S^(2n+1) extends to
such a fibration of all of S^(2n+1), a result previously known only for S^3 .
