### 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).