Fiberwise homogeneous fibrations of the 3-dimensional space forms by geodesics

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. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. 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. By contrast, we show in another paper that the only fiberwise homogeneous fibrations of the round 3-sphere by geodesics (great circles) are the Hopf fibrations. More generally, we show in that same paper that the Hopf fibrations in all dimensions are characterized, among fibrations of round spheres by smooth subspheres, by their fiberwise homogeneity.