On three-manifolds admitting a Killing vector field

A Killing submersion is a Riemannian submersion, from an orientable three-manifold to an orientable surface, whose fibers are the integral curves of a Killing vector field. Indeed, any three-manifold with a Killing vector field can be locally endowed with a Killing submersion structure away from zeroes of the vector field. The first part of this talk will be devoted to discuss local and global classification results for Killing submersions by prescribing an arbitrary base surface and two arbitrary geometric functions, namely the bundle curvature and the length of the Killing vector field. In this discussion, we will assume that the total space is either Riemannian or Lorentzian. Moreover, we will show that the only Killing submersions with Killing vector field of constant length in which the total space is homogeneous are the Bianchi-Cartan-Vranceanu spaces, i.e., the so-called $\mathrm{E}(\kappa,\tau)$-spaces. Time permitting, we will also discuss a geometric duality for graphical spacelike surfaces between Riemannian and Lorentzian Killing submersions that swaps the mean curvature of the surface and the bundle curvature of the ambient space. This talk is partially based on joint works with Hojoo Lee and Ana M. Lerma.