On the classification of Killing submersions and their isometries

José M. Manzano
2014 Pacific Journal of Mathematics  
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
more » ... irection. As a consequence, we prove that the only simply-connected homogeneous 3-manifolds which admit a structure of Killing submersion are the E(\kappa,\tau)-spaces, whose isometry group has dimension at least 4.
doi:10.2140/pjm.2014.270.367 fatcat:quqylz6tmrffji4plar364gfni