On the structure of hypersurfaces in Hn×R with finite strong total curvature

We prove that if X : Mn → Hn×R, n≥3, is an orientable, complete immersion with finite strong total curvature, then X is proper and M is diffeomorphic to a compact manifold M minus a finite number of points q1, ... qk. Adding some extra hypothesis, including Hr = 0, where Hr is a higher order mean curvature, we obtain more information about the geometry of a neighbourhood of each puncture. The reader will also find in this paper a classification result for the hypersurfaces of Hn × R which satisfy Hr = 0 and are invariant by hyperbolic translations and a maximum principle in a half-space for these hypersurfaces.