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Zbl 1041.58017
Izumiya, Shyuichi; Pei, Donghe; Sano, Takasi
Singularities of hyperbolic Gauss maps. (English)
[J] Proc. Lond. Math. Soc., III. Ser. 86, No. 2, 485-512 (2003). ISSN 0024-6115; ISSN 1460-244X/e

The authors study the differential geometry of hypersurfaces in hyperbolic space. They reformulate the questions and adapt the methods coming from the classical theory of Gauss map of a surface in Euclidean 3-space. They study singularities of hyperbolic Gauss map by adapting the hyperboloid in Minkowski space as the model of hyperbolic space, and using the local parametrization of the hypersurface. Redefining of the notion of the light-cone normal and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space they study two hyperbolic invariants; the hyperbolic Gauss-Kronecker curvature and the hyperbolic mean curvature. The theory of Legendrian singularities is applied to study hyperbolic Gauss indicatrices in connection to the classical property of the Gauss-map to be a Lagrangian map.
[Stanislaw Janeczko (Warszawa)]

MSC 2000:: *58K30 Global theory
53C40 Submanifolds (differential geometry)

Keywords: Gauss map; hyperbolic space; generic singularities; Legendrian map

Cited in: Zbl 1205.53065 Zbl 1113.53011 Zbl 1074.53045

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