The singularities of null surfaces in Anti de Sitter 3-space. (English) Zbl 1185.53017

Summary: We study the geometric properties of degenerate surfaces, which are called AdS null surfaces, in Anti de Sitter 3-space from a contact viewpoint. These surfaces are associated to space-like curves in Anti de Sitter 3-space. We define a map which is called the torus Gauss image. We also define two families of functions and use them to investigate the singularities of AdS null surfaces and torus Gauss images as applications of singularity theory of functions.


53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
58K60 Deformation of singularities
83C15 Exact solutions to problems in general relativity and gravitational theory
Full Text: DOI


[1] Chen, L., On spacelike surfaces in Anti de Sitter 3-space from the contact viewpoint, Hokkaido Math. J., 38, 701-720 (2009) · Zbl 1180.53018
[3] Izumiya, S.; Pei, D.-H.; Sano, T., The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space, Glasg. Math. J., 42, 75-89 (2000) · Zbl 0970.53015
[4] Fusho, T.; Izumiya, S., Lightlike surfaces of spacelike curves in de Sitter 3-space, J. Geom., 88, 19-29 (2008) · Zbl 1136.53008
[5] Bruce, J. W.; Giblin, P. J., Curves and Singularities (1992), Cambridge Univ. Press · Zbl 0770.53002
[6] Wassermann, G., Stability of caustics, Math. Ann., 2210, 443-450 (1975)
[7] Martinet, J., Singularities of Smooth Functions and Maps, London Math. Soc. Lecture Note Ser., vol. 58 (1982), Cambridge Univ. Press · Zbl 0522.58006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.