BSU Bulletin. Mathematics, Informatics
Bibliographic description:
Complexes in three-dimensional quasi-hyperbolic space // BSU Bulletin. Mathematics, Informatics. - 2016. №1. . - С. 9-15.
Title:
Complexes in three-dimensional quasi-hyperbolic space
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Annotation:
In the article the canonical frame of a complex is constructed. This frame is
geometrically characterized by the fact that in normal correlation the points A0
and A1 (centers of complex-ray) correspond to the planes (A0 A1 A2) and (A0 A1 A3), which are polar conjugated with respect to the absolute and cross absolute line to the points
A2 and A3 . The theorem of existence is proved. We have given the geometric characteristics of the complex invariants using three simple ruled surfaces (central surface and two central torses) belonging to the complex.
Two main quadratic forms of the complex have been obtained. The ruled surfaces conjugated with respect to the first quadratic form are characterized by the harmonic conjugation of their adherent points. The surfaces conjugated with respect to the second quadratic form are characterized by the harmonic conjugation of the adherent points of one of them with the symmetry points of the other.
We have obtained the equation of inflectional centers of the complex generatrices, the conditions characterizing the linear complex, and found some spe- cial classes of the complexes.
Keywords:
non-Euclidean space, quasi-hyperbolic space, absolute, com- plex, frame, normal correlation, invariants.
List of references:
1. Shcherbakov R. N. Osnovy metoda vneshnikh form i lineichatoi different- sial'noi geometrii [Fundamentals of the Method of External Forms and Ruled Differential Geometry]. Tomsk: Tomsk State University Publ., 1973. 236 p.
2. Tsyrenova V. B. Kompleksy v trekhmernom kvaziellipticheskom pro- stranstve [Complexes in Three-Dimensional Quasi-Elliptic Space]. Geometri- cheskii sbornik – Geometric Collection. 1985. No. 25. Pp. 91-100.
3. Tsyrenova V. B., Proskuryakova I. V. Kompleksy v trekhmernom kvazi- giperbolicheskom prostranstve [Complexes in Three-Dimensional Quasi- Hyperbolic Space]. Vestnik Buryatskogo gosudarstvennogo universiteta. Ma- tematika, informatika – Bulletin of Buryat State University. Mathematics, In- formatics. 2011. No. 1. Pp. 92–94.