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.
Shyuichi IZUMIYA. María del Carmen ROMERO FUSTER. "The horospherical Gauss-Bonnet type theorem in hyperbolic space." J. Math. Soc. Japan 58 (4) 965 - 984, October, 2006. https://doi.org/10.2969/jmsj/1179759532