| In the first part of this talk we will discuss the existence and uniqueness of Riemannian submersions from 3-manifolds to surfaces, whose fibers are the integral curves of a unit Killing vector field. This will be accomplished by considering a geometric function called bundle curvature. The second part of the talk will be devoted to study surfaces in the total space of such a submersion which are transverse to the Killing vector field. We will focus on a generalization to this spaces of the classical Calabi correspondence between minimal surfaces in the Euclidean 3-space and maximal surfaces in the Minkowski 3-space. Indeed we shall see that the correspondence swaps the mean curvature of the graph and the bundle curvature of the space, and we will give some applications. |