Horo-flat surfaces along cuspidal edges in the hyperbolic space

Shyuichi Izumiya, Maria Carmen Romero-Fuster, Kentaro Saji, and Masatomo Takahashi

Journal of Singularities
volume 22 (2020), 40-58

Received: 26 February 2019. Received in revised form: 8 June 2020.

DOI: 10.5427/jsing.2020.22d

Add a reference to this article to your citeulike library.

Abstract:

There are two important classes of surfaces in the hyperbolic space. One of class consists of extrinsic flat surfaces, which is an analogous notion to developable surfaces in the Euclidean space. Another class consists of horo-flat surfaces, which are given by one-parameter families of horocycles. We use the Legendrian dualities between hyperbolic space, de Sitter space and the lightcone in the Lorentz-Minkowski 4-space in order to study the geometry of flat surfaces defined along the singular set of a cuspidal edge in the hyperbolic space. Such flat surfaces can be considered as flat approximations of the cuspidal edge. We investigate the geometrical properties of a cuspidal edge in terms of the special properties of its flat approximations.

2010 Mathematical Subject Classification:

Primary 57R45; Secondary 58Kxx

Key words and phrases:

cuspidal edges, flat approximations, curves on surfaces, Darboux frame, horo-flat surfaces

Author(s) information:

Shyuichi Izumiya
Department of Mathematics
Hokkaido University
Sapporo 060-0810, Japan
email: izumiya@math.sci.hokudai.ac.jp

Maria del Carmen Romero Fuster
Department de Geometria i Topologia
Universitat de Valencia
46100 Valencia, Spain
email: Carmen.Romero@uv.es

Kentaro Saji
Department of Mathematics
Kobe University
Rokko 1-1, Nada, Kobe 657-8501, Japan
email: saji@math.kobe-u.ac.jp

Masatomo Takahashi
Muroran Institute of Technology
Muroran 050-8585, Japan
email: masatomo@mmm.muroran-it.ac.jp