The horospherical Gauss-Bonnet type theorem in hyperbolic space (Q861745)

In the very interesting paper the authors introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space and show the following main-theorem: If \(M\) is a closed orientable even-dimensional hypersurface in hyperbolic \(n\)-space, then \[ \int_M\widetilde K_h\,dv_M= {1\over 2}\gamma_{n-1}\chi(M), \] where \(\chi(M)\) is the Euler characteristic of \(M\), \(dv_M\) is the volume form of \(M\), \(\widetilde K_h\) is the horospherical Gauss-Kronecker-curvature and the constant \(\gamma_{n-1}\) is the volume of the unit \((n-1)\)-sphere \(\mathbb{S}^{n-1}\). Furthermore, they show that the only totally umbilic hypersurfaces with vanishing curvatures are horospheres. Finally, the authors study in this context surfaces in hyperbolic 3-space. The paper gives nice connexions with results of T. Kobayashi and C. L. Epstein.