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Locally convex surfaces immersed in a Killing submersion

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.