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 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.