  We generalize Hadamard-Stoker-Currier Theorems for surfaces immersed in a
Killing submersion over a strictly Hadamard surface whose fibers are the
trajectories of a unit Killing field. We prove that every complete surface
whose principal curvatures are greater than a certain function (depending on
the ambient manifold) at each point, must be properly embedded, homeomorphic to
the sphere or to the plane and, in the latter case, we study the behavior of
the end.
