Geodesic
Classification of tetrahedrons with not hyperbolic sides in a hyperbolic space of positive curvature
Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 208-221.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: hyperbolic space $\widehat{H}^3$ of positive curvature, classification of tetrahedrons of a hyperbolic space of positive curvature. 
@article{CHEB_2015_16_2_a12,
     author = {L. N. Romakina},
     title = {Classification of tetrahedrons with not hyperbolic sides in a hyperbolic space of positive curvature},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {208--221},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2015_16_2_a12/}
}
TY  - JOUR
AU  - L. N. Romakina
TI  - Classification of tetrahedrons with not hyperbolic sides in a hyperbolic space of positive curvature
JO  - Čebyševskij sbornik
PY  - 2015
SP  - 208
EP  - 221
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2015_16_2_a12/
LA  - ru
ID  - CHEB_2015_16_2_a12
ER  - 
%0 Journal Article
%A L. N. Romakina
%T Classification of tetrahedrons with not hyperbolic sides in a hyperbolic space of positive curvature
%J Čebyševskij sbornik
%D 2015
%P 208-221
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2015_16_2_a12/
%G ru
%F CHEB_2015_16_2_a12
L. N. Romakina. Classification of tetrahedrons with not hyperbolic sides in a hyperbolic space of positive curvature. Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 208-221. http://geodesic.mathdoc.fr/item/CHEB_2015_16_2_a12/

[1] Klein F., Non-Euclidean geomtry, Joint Scientific and Technical Publishing House, M.–L., 1936, 356 pp.

[2] Efimov N. V., The highest geometry, Nauka, M., 1971, 576 pp. | MR

[3] Norden A. P., The spaces of affine connectivity, Nauka, M., 1976, 432 pp. | MR

[4] Rozenfeld B. A., Non-Euclidean geomtry, State Publ. of Technical and Theoretical Literature, M., 1955, 744 pp. | MR

[5] Rozenfeld B. A., Non-Euclidean spaces, Nauka, M., 1969, 548 pp. | MR

[6] Rozenfeld B. A., Zamakhovskii M. P., Geometry of groups of Lie. Symmetric, parabolic and periodic spaces, MCCME, M., 2003, 560 pp.

[7] Ponarin J. P., Geometry with affine base, Publishing house of the Kirov state pedagogical university, Kirov, 1991, 121 pp.

[8] Yaglom I. M., Principle of relativity Galilee and non-Euclidean geometry, Editorial URSS, M., 2004, 303 pp. | MR

[9] Romakina L. N., Geometry of the hyperbolic plane of positive curvature, in 4 pt., v. 1, Trigonometry, Saratov Univ. Press, Saratov, 2013, 244 pp.

[10] Romakina L. N., Geometry of the hyperbolic plane of positive curvature, in 4 pt., v. 2, Transformations and simple splittings, Saratov Univ. Press, Saratov, 2013, 274 pp.

[11] Romakina L. N., Geometry of the coeuclidean and copseudoeuclidean planes, Publishing House «Scientific Book», Saratov, 2008, 279 pp.

[12] Romakina L. N., “Analogs of a formula of Lobachevsky for angle of parallelism on the hyperbolic plane of positive curvature”, Siberian Electronic Mathematical Reports, 10 (2013), 393–407 (in Russian) http://semr.math.nsc.ru | MR

[13] Romakina L. N., “Concerning parabolic polyhedrons of copseudoeuclidean space”, Bulletin of “KSPU”, 2013, no. 1, 201–206

[14] Aleksandrov A. D., Convex polyhedrons, State Publ. of Technical and Theoretical Literature, M., 1950, 429 pp. | MR

[15] Ashkinuze V. G., “Polygons and polyhedrons”, Encyclopedia of elementary mathematics, v. IV, Geometry, Physical and Mathematical State Publishing House, M., 1963