Geometric and analytic interpretation of orthoscheme and Lambert cube in extended hyperbolic space (Q2868504)

The authors prove theorems regarding the volume formula in \(3\)-dimensional extended hyperbolic space (in the sense of the authors' [Geom. Dedicata 161, 129--155 (2012; Zbl 1260.51006)]. Two sample theorems are: (i) Theorem 3.2: For a given tetrahedron inside \({\mathbb H}^3\) with three ideal vertices, if we move the non-ideal vertex out continuously from inside to outside of \({\mathbb H}^3\), then the volume formula is expressed by an analytic multi-valued function. The analytic function will take a unique value for a contour choice, and only when the moving vertex is on the ideal boundary \(\partial {\mathbb H}^3\), the function is not analytic but continuous at that point. (ii) Theorem 4.8: For a hyperbolic tetrahedron with six dihedral angle variables, if we move vertices out continuously from inside to outside of \({\mathbb H}^3\) keeping every face of the tetrahedron intersecting the hyperbolic space \({\mathbb H}^3\), then the volume formula is expressed by the same analytic multi-valued function. This analytic function takes a unique value for a clockwise (counterclockwise resp.) contour choice, and this choice coincides with the volume of the tetrahedron of the fixed extended hyperbolic space defined from the clockwise (counterclockwise resp.) contour. When the moving vertex is on the ideal boundary \(\partial {\mathbb H}^3\), the volume function is not analytic but is continuous at the point. In particular, at the edge-tangent case (i.e., when the dihedral angle at the edge is \(0\)) the volume function of dihedral angle variables is analytic. NEWLINENEWLINETheorem 3.2 is proved by elementary geometric methods, and the volume formulas obtained provide \textit{more geometrico} alternatives to results in [\textit{R. Kellerhals}, Math. Ann. 285, No. 4, 541--569 (1989; Zbl 0664.51012) and \textit{A. Ushijima}, Math. Appl., Springer 581, 249--265 (2006; Zbl 1096.52006)].