Dualities and envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐spaces

Abstract

We consider envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one-parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one-parameter families of frontals in hyperbolic and de Sitter 2-space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes. K E Y W O R D Sde Sitter 2-space, envelope, frontal, hyperbolic 2-space, Legendrian duality M S C ( 2 0 1 0 ) 53A35, 53B30, 58K05

Singularities of Frontal Surfaces

“…There has been a great interest for frontals in the last decades, specially in the C ∞ real category and looking at differential geometric properties. The fact that you have a well defined tangent plane everywhere provides a nice starting point if you want to extend things like first or second fundamental forms, curvatures, etc to submanifolds with singularities (see for instance [5,18,27,29,30]).…”