A Killing submersion is a Riemannian submersion from an orientable
-manifold
to an orientable surface whose fibers are the integral curves of a unit Killing vector field in
the
-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
-manifolds
which admit a structure of Killing submersion are the
-spaces, whose isometry
group has dimension at least
.
Keywords
unit Killing vector field, Riemannian submersions,
homogeneous spaces
