Analytical method for study of polyhedra and orbifolds in three dimensional spaces of constant curvature

The work is devoted to analytical methods for the study of geometric properties for some classes of three-dimensional polyhedra, as well as orbifolds in spaces of constant curvature. In the first part of the work, we consider hyperbolic tetrahedra and octahedra with symmetries. Necessary and sufficient conditions of existence have been established for them; obtaining relations between the lengths of the edges and dihedral angles in the form of cosine laws; explicit integral formulas for hyperbolic volumes are obtained. The second part of the work is devoted to the study of antiprisms in spaces of constant curvature. Criteria for existence are established, relations between the lengths of edges and dihedral angles are found, and exact formulas for the volumes of antiprisms in Euclidean, spherical, and hyperbolic spaces are obtained. In the third part of the thesis, it is proved that all three-dimensional closed orientable Euclidean manifolds, except for the three-dimensional torus, are hyperelliptic, that is, they are two-sheeted coverings of the three-dimensional sphere branched over knots and links. A one-to-one correspondence between the indicated manifolds and Euclidean orbifolds is established. It is shown how the hyperelliptic involution acts on these manifolds.