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 S n c ×R and H n c ×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 ≥ 4 we give examples of pseudo-parallel surfaces with non vanishing normal curvature.