The horospherical Gauss-Bonnet type theorem in hyperbolic space

Abstract

We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space and show that totally umbilic hypersurfaces with vanishing cur- vatures 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. + (i1) by using the model in Minkowski space. We introduced the notion of hyperbolic Gauss indicatrices slightly modified the definition of hyperbolic Gauss maps. The notion of hyperbolic indicatrices is independent of the choice of the model of hyperbolic space. Using the hyperbolic Gauss indicatrix, we defined the principal hyperbolic curvatures ¯ • § and the hyperbolic Gauss-Kronecker curvature K § h by exactly the same way as the definition of those of classical Gaussian dif- ferential geometry in Euclidean space. Totally umbilic hypersurfaces with respect to the above curvatures are equidistant hypersurfaces, hyperspheres or hyperhorospheres which are called model hypersurfaces in hyperbolic space. The hyperbolic Gauss-Kronecker curvature is a hyperbolic invariant which describes the contact of hypersurfaces with such model hypersurfaces. We remark that Kobayashi (13), (14) had already defined the notion of hyperbolic Gauss-Kronecker curvature under a dierent framework and studied some basic properties of it from the view point of the theory of Fourier transformations. In this paper we introduce the principal horospherical curvature e § (cf., x3). This new curvature is not a hyperbolic invariant but an SO(n)-invariant, where we consider the canonical SO(n)-subgroup in the group of hyperbolic motions. However, we can show that e § (p) is invariant under hyperbolic motions if and only if e § (p) = 0. We can