Classification of tetrahedrons with not hyperbolic sides in a hyperbolic space of positive curvature

L. N. Romakina

Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 208-221

Résumé

The work contains the research of tetrahedrons of a hyperbolic space $\widehat {H} ^3$ of positive curvature. The space $\widehat {H} ^3$ is realized by on external domain of projective three-dimensional space with respect to the oval hyperquadric, т. e. on ideal domain of the Lobachevskii space. All lines of the space $\widehat{H}^3$ on existence of the common points with the absolute can be elliptic, parabolic or hyperbolic. All planes of the space $\widehat{H}^3$ depending on position with respect to the absolute belong to the three types: elliptic, coeuclidean and hyperbolic of positive curvature. The angles of the elliptic plane of one type. The angles of the coeuclidean plane of three types. The angles of the hyperbolic plane of positive curvature of fifteen types. Also all dihedral angles of the space $\widehat{H}^3$ belong to fifteen types. Various sets of types of sides define in the space $\widehat {H}^3$ fifteen types of tetrahedrons. In work the classification of tetrahedrons with not hyperbolic sides is carried out. All such tetrahedrons belong to five types. It is proved that each edge of a tetrahedron with not hyperbolic sides belongs to the elliptic line. The elliptic line is the closed line in the space $\widehat{H}^3$. In further classification of tetrahedrons with not hyperbolic sides we use concept $\alpha$-sides of the tetrahedron. In the space $\widehat{H}^3$ the cone of tangents to absolute oval hyperquadric is connected with each point. This cone we called a light cone of a point. The curve of crossing of the side plane with a light cone of tetrahedron top, opposite to this side, is called the light curve of a side of a tetrahedron. The tetrahedron side in the space $\widehat{H}^3$ is called a $\alpha$-side if it contains completely the light curve. The following theorem is proved. The tetrahedron with not hyperbolic sides in the space $\widehat{H}^3$ or doesn't contain the $\alpha$-sides, or supports one $\alpha$-side, or all its sides are the $\alpha$-sides. Quantity $\alpha$-sides and their types define classes and sorts of the tetrahedrons with not hyperbolic sides. In work the types of dihedral angles in a tetrahedron of each class (sort) are established. Bibliography: 4 titles.

Détail

Keywords: hyperbolic space $\widehat{H}^3$ of positive curvature, classification of tetrahedrons of a hyperbolic space of positive curvature.