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

E N Sinyukova,O L Chepok

doi:10.13189/ms.2020.080609

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 <img src=image/13417732_01.gif> that is considered as a subset of Euclidean space <img src=image/13417732_02.gif>, 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.

Highlights

We will consider as a differentiable manifold Mn of the class Cr, n, r ∈ N, r > 1, a Hausdorff topological space that satisfies the second axiom of countability, every point of which has a neighborhood that is homeomorphic to some domain of the space Rn
Any two coordinate neighborhoods are Cr-related in the sense that appeared in the case of their nonempty intersection functions of transformation of the one coordinate system to the other, are the smooth functions of the class C r
On the ground of the sphere Sn that is considered as a subspace of Euclidean space En+1, n ∈ N, n ≥ 1, an example of the global solution of the new form of the basic equations of the theory of geodesic mappings of Riemannian space is built

Summary

Introduction

We will consider as a differentiable manifold Mn of the class Cr, n, r ∈ N, r > 1, a Hausdorff topological space that satisfies the second axiom of countability, every point of which has a neighborhood that is homeomorphic to some domain of the space Rn. There exist wide classes of Riemannian spaces that locally admit non-trivial (different from the affine) geodesic mappings and, at the same time, don’t admit such mappings “on the whole” ([5, 6], for example).

Results

Conclusion