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SINGULARITIES OF HYPERBOLIC GAUSS MAPS

Published online by Cambridge University Press:  06 March 2003

SHYUICHI IZUMIYA ,
DONGHE PEI  and
TAKASI SANO
SHYUICHI IZUMIYA
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan. [email protected]
DONGHE PEI
Affiliation:
Department of Mathematics, North East Normal University, Chang Chun 130024, P. R. China. [email protected]
TAKASI SANO
Affiliation:
Faculty of Engineering, Hokkaigakuen University, Sapporo 062-8605, Japan. [email protected]
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Abstract

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.

2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

Keywords

hyperbolic spacehypersurfaceshyperbolic Gauss mapssingularities
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 86 , Issue 2 , March 2003 , pp. 485 - 512
Copyright
2003 London Mathematical Society

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