Lines on the surface in the quasi-hiperbolic space
- DOI
- 10.5922/0321-4796-2020-51-14
- Pages
- 123-134
Abstract
Quasi-hyperbolic spaces are projective spaces with decaying absolute. This work is a continuation of the author's work [7], in which surfaces in one of these spaces are examined by methods of external forms and a moving frame. The semi-Chebyshev and Chebyshev networks of lines on the surface in quasi-hyperbolic space are considered. In this paper we use the definition of parallel transfer adopted in [6]. By analogy with Euclidean geometry, the semi-Chebyshev network of lines on the surface is the network in which the tangents to the lines of one family are carried parallel along the lines of another family. A Chebyshev network is a network in which tangents to the lines of each family are carried parallel along the lines of another family.
We proved three theorems. In Theorem 1, we obtain a natural equation for non-geodesic lines that are part of a conjugate semi-Chebyshev network on the surface so that tangents to lines of another family are transferred in parallel along them. In Theorem 2, the natural equation of non-geodesic lines in the Chebyshev network is obtained. In Theorem 3 we prove that conjugate Chebyshev networks, one family of which is neither geodesic lines, nor Euclidean sections, exist on surfaces with the latitude of four functions of one argument.
Reference
1. Rosenfeld, B. A.: Non-Euclidean spaces. Moscow (1969).
2. Scherbakov, R. N.: Course of affine and projective differential geometry. Tomsk (1960).
3. Guryeva, V. P., Abdurakhmanova, Kh. K.: On the theory of surfaces in three-dimensional quasi-elliptic and quasi-hyperbolic spaces. Geom. Sb. Tomsk. 17, 132—139 (1976).
4. Slobodskoy, V. I.: The theory of surfaces in three-dimensional quasi-hyperbolic space. Geom. Sb. Tomsk. 21, 55—67 (1980).
5. Tsyrenova, V. B., Scherbakov, R. N.: Fundamentals of the theory of surfaces of three-dimensional quasielliptic space. Geom. Sb. Tomsk. 15, 183—204 (1975).
6. Tsyrenova, V. B.: On the theory of surfaces in quasielliptic space. Geom. Sb. Tomsk. 19, 96—108 (1978).
7. Tsyrenova, V. B.: Surfaces in quasi-hyperbolic space . Geometry of manifolds and its applications: materials of the Fifth Scientific Conference. Ulan-Ude. 56—60 (2018).