Fiberwise homogeneous geodesic foliations of hyperbolic and Euclidean $3$–spaces

Abstract

A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. In this paper, we describe all the fiberwise homogeneous fibrations of Euclidean and hyperbolic $3$–space by geodesics. Our main result is that, up to fiber-preserving isometries, there is precisely a one-parameter family of such fibrations of Euclidean $3$–space, and a two-parameter family in hyperbolic $3$–space.