Journal of the Mathematical Society of Japan

The horospherical Gauss-Bonnet type theorem in hyperbolic space

Shyuichi IZUMIYA and María del Carmen ROMERO FUSTER

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We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space andshow that totally umbilic hypersurfaces with vanishing curvatures are only horospheres. We also show that the Gauss-Bonnet type theorem holds for the horospherical Gauss-Kronecker curvature of a closed orientable even dimensional hypersurface in hyperbolic space.

Article information

J. Math. Soc. Japan, Volume 58, Number 4 (2006), 965-984.

First available in Project Euclid: 21 May 2007

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Zentralblatt MATH identifier

Primary: 53A35: Non-Euclidean differential geometry
Secondary: 53A05: Surfaces in Euclidean space 58C27

hyperbolic space hypersurfaces hyperbolic Gauss maps horospherical geometry Gauss-Bonnet type theorem


IZUMIYA, Shyuichi; ROMERO FUSTER, María del Carmen. The horospherical Gauss-Bonnet type theorem in hyperbolic space. J. Math. Soc. Japan 58 (2006), no. 4, 965--984. doi:10.2969/jmsj/1179759532.

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