Pseudo-parallel surfaces of $$\mathbb {S}_c^n \times \mathbb {R}$$ S c n × R and $$\mathbb {H}_c^n \times \mathbb {R}$$ H c n × R

Abstract

In this work we give a characterization of pseudo-parallel surfaces in $$\mathbb {S}_c^n \times \mathbb {R}$$ and $$\mathbb {H}_c^n\times \mathbb {R}$$ , extending an analogous result by Asperti-Lobos-Mercuri for the pseudo-parallel case in space forms. Moreover, when $$n=3$$ , we prove that any pseudo-parallel surface has flat normal bundle. We also give examples of pseudo-parallel surfaces which are neither semi-parallel nor pseudo-parallel surfaces in a slice. Finally, when $$n\ge 4$$ we give examples of pseudo-parallel surfaces with non vanishing normal curvature.