Volumes of polyhedra in non-Euclidean spaces of constant curvature

V. A. Krasnov

Contemporary Mathematics. Fundamental Directions, Algebra, Geometry, and Topology, Tome 66 (2020) no. 4, pp. 558-679

Résumé

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. Deriving volume formulas for $3$-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevsky who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute). In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in $3$-dimensional spherical and hyperbolic spaces of constant curvature $K=1$ and $K=-1$, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in hyperbolic space (described by the Poincaré model in upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.