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
Publication
Received: 24 November 2014
Revised: 6 January 2015
Accepted: 20 February 2015
Published: 10 December 2015
Authors
 Haggai Nuchi Department of Mathematics University of Toronto Bahen Centre 40 George St, Room 6290 Toronto M5S 2E4 Canada