Locally convex surfaces immersed in a Killing submersion

TL;DR

This paper extends classical theorems about convex surfaces to those immersed in a Killing submersion, establishing conditions under which such surfaces are properly embedded and characterizing their topology and ends.

Contribution

It generalizes Hadamard-Stoker-Currier theorems to surfaces in Killing submersions, providing new curvature conditions for proper embedding and topological classification.

Findings

Complete surfaces with principal curvatures above a certain threshold are properly embedded.

Such surfaces are topologically spheres or planes.

The behavior of ends of non-compact surfaces is analyzed.

Abstract

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.