### Journals Information

**
Mathematics and Statistics Vol. 8(6), pp. 693 - 698 DOI: 10.13189/ms.2020.080609 Reprint (PDF) (638Kb) **

## On Some Global Solution of the Basic Equations in the Geodesic Mappings' Theory of Riemannian Spaces

**E. N. Sinyukova ^{1}^{,*}, O. L. Chepok ^{2}**

^{1}Department of Mathematics and Statistics, South Ukrainian National Pedagogical University named after K. D. Ushynsky, Odessa, 65026, Ukraine

^{2}Department of Physics, South Ukrainian National Pedagogical University named after K. D. Ushynsky, Odessa, 65026, Ukraine

**ABSTRACT**

It is well known that concepts of a geodesic line and a geodesic mapping are among the most fundamental concepts of classical theory of Riemannian spaces. In geometry, concept of Riemannian space has been formed as a generalization of the concept of a smooth surface in a three-dimensional Euclidean space. It has turned out to be possible to extend to Riemannian space the concept of a geodesic point of a curve and to represent a geodesic line of Riemannian space as a curve that consists exclusively of geodesic points. The fact has allowed understanding not only the local but also the global character of basic equations of geodesic mappings' theory of Riemannian spaces that have been originally received as a result of local investigations. An example of the global solution of the so-called new form of basic equations in the theory of geodesic mappings of Riemannian spaces is built in the article. Sphere that is considered as a subset of Euclidean space , forms its topological background. Investigations are based on the concept of equidistant Riemannian space. They are carried out according to the atlas that consists of two charts, obtained with the help of a stereographic projection.

**KEYWORDS**

Riemannian Space, Geodesic Line, Local Geodesic Mapping, Global Geodesic Mapping, Equidistant Riemannian Space, Sphere

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] E. N. Sinyukova , O. L. Chepok , "On Some Global Solution of the Basic Equations in the Geodesic Mappings' Theory of Riemannian Spaces," Mathematics and Statistics, Vol. 8, No. 6, pp. 693 - 698, 2020. DOI: 10.13189/ms.2020.080609.

(b). APA Format:

E. N. Sinyukova , O. L. Chepok (2020). On Some Global Solution of the Basic Equations in the Geodesic Mappings' Theory of Riemannian Spaces. Mathematics and Statistics, 8(6), 693 - 698. DOI: 10.13189/ms.2020.080609.