Differential Geometry of Manifolds

2020 №51

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Lines on the surface in the quasi-hiperbolic space



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.


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