The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and ``flat'' spacelike surfaces

Shyuichi Izumiya and Kentaro Saji

Journal of Singularities
volume 2 (2010), 92-127
Proceedings of Singularities in Aarhus, August 2009

Received: 28 September 2009

DOI: 10.5427/jsing.2010.2g

Version with corrections: received on December 31, 2010.
Revisions are indicated in red in the pdf. References should be made to the original, published article.

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Using the Legendrian dualities between surfaces in pseudo-spheres in Lorentz-Minkowski 4-space, we study various kind of flat surfaces in pseudo-spheres. We consider a surface in the pseudo-sphere and its dual surface. Flatness of a surface is defined by the degeneracy of the dual surface similar to the case for the Gauss map of a flat surface in the Euclidean space. We study singularities of these flat surfaces and dualities of singularities.


Legendrian duality, Horo-spherical geometry, Horo-flat surfaces, Singularities, Lorentz-Minkowski space, Pseudo-spheres

Mathematical Subject Classification:

53A35, 57R45, 53C42

Author(s) information:

Department of Mathematics, Faculty of Science Department of Mathematics, Faculty of Education
Hokkaido University Gifu University
Sapporo 060-0810, Japan Yanagido 1-1, Gifu, 501-1193, Japan
email: email: