# On the classification of Killing submersions and their isometries

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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\kappa,\tau-spaces, whose isometry group has dimension at least 4.

Author: José M. Manzano

Source: https://archive.org/