On the classification of Killing submersions and their isometries

TL;DR

This paper classifies Killing submersions from 3-manifolds to surfaces, describes their isometries, and identifies the specific homogeneous 3-manifolds that admit such structures, notably the E(, au)-spaces.

Contribution

It provides a complete classification of Killing submersions over simply-connected surfaces and characterizes their isometry groups, linking them to E(, au)-spaces.

Findings

Classified all Killing submersions over simply-connected surfaces.

Explicit models for many Killing submersions were constructed.

Proved that only E(, au)-spaces among simply-connected homogeneous 3-manifolds admit Killing submersion structures.

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