On the classification of killing submersions and their isometries

Abstract

A Killing submersion is a Riemannian submersion from an orientable 3- manifold to an orientable surface whose fibers are the integral curves of a unit Killing vector field in the 3-manifold. We classify all Killing submersions over simply connected Riemannian surfaces and give explicit models for many Killing submersions, including those over simply connected constant Gaussian curvature surfaces. We also fully describe the isometries of the total space preserving the vertical direction. As a consequence, we prove that the only simply connected homogeneous 3-manifolds which admit a structure of Killing submersion are the E.(κ, τ)-spaces, whose isometry group has dimension at least 4. © 2014 Mathematical Sciences Publishers.