Lightlike tangent developables in de Sitter 3-space

Abstract Developable surfaces are the special ruled surfaces where the Gaussian curvature of each point vanishes. Tangent developable is the most interesting surface among three fundamental types of developable surfaces. This paper investigates lightlike tangent developable generated by a lightlike base curve in de Sitter 3-space. We first give the topological classification of the lightlike curves which lie in de Sitter 3-space by using the theory of finite determinacy. Furthermore, we establish the relationships between lightlike tangent developables and topological types of lightlike curves, and we give the classification of singularities of lightlike tangent developables from the viewpoint of Legendrian singularity theory. Finally, we provide several examples to illustrate our theoretical results.

[1]  S. Izumiya,et al.  Lightlike surfaces of spacelike curves in de Sitter 3-space , 2006 .

[2]  Singularities of null developable of timelike curve that lies on nullcone in semi-Euclidean 3-space with index 2 , 2013 .

[3]  D. Pei,et al.  Pedal curves of fronts in the sphere , 2016 .

[4]  J. P. Cleave,et al.  The form of the tangent-developable at points of zero torsion on space curves , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  D. Pei,et al.  Pedal curves of frontals in the Euclidean plane , 2018 .

[6]  Geometry of 1-lightlike submanifolds in anti-de Sitter n-space , 2013, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  G. Ishikawa.  Topological Classification of the Tangent Developables of Space Curves , 1996 .

[8]  J. Ballesteros.  On the number of triple points of the tangent developable , 1993 .

[9]  Singularities of ruled null surfaces of the principal normal indicatrix to a null Cartan curve in de Sitter 3-space , 2010 .

[10]  A. Plessis,et al.  More on the determinacy of smooth map-germs , 1982 .

[11]  D. Mond.  On the tangent developable of a space curve , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  T. Zhao,et al.  Slant helix of order n and sequence of Darboux developables of principal‐directional curves , 2020, Mathematical Methods in the Applied Sciences.

[13]  Yushu Zhu,et al.  Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space , 2020 .

[14]  D. Pei,et al.  Envelopes of legendre curves in the unit spherical bundle over the unit sphere , 2018 .

[15]  Charles Terence Clegg Wall,et al.  Finite Determinacy of Smooth Map‐Germs , 1981 .

[16]  D. Mond.  SINGULARITIES OF THE TANGENT DEVELOPABLE SURFACE OF A SPACE CURVE , 1989 .

[17]  D. Pei,et al.  Evolutes of dual spherical curves for ruled surfaces , 2016 .

[18]  Feng Ye.  Semi-Riemannian Geometry , 2011 .

[19]  S. Izumiya,et al.  LIGHTLIKE DEVELOPABLES IN MINKOWSKI 3-SPACE , 2006 .

[20]  Shyuichi Izumiya,et al.  New special curves and developable surfaces , 2002 .

[21]  Gaussian surfaces and nullcone dual surfaces of null curves in a three-dimensional nullcone with index 2 , 2013 .

[22]  Yanlin Li,et al.  Tangent developables and Darboux developables of framed curves , 2020 .