#### Volume 15, issue 5 (2015)

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

### Haggai Nuchi

Algebraic & Geometric Topology 15 (2015) 3057–3068
##### 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.

##### Keywords
homogeneous, geodesic, foliation, hyperbolic, Euclidean, 3-space
##### Mathematical Subject Classification 2010
Primary: 53C12
Secondary: 57R30, 57M60, 57S20