NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF A QUASIO-DOUBLE LINE OF A THE PARTIAL MAPPING IN SPACE E5

Abstract

It is considered the problem related to partial mapping of 5-dimensional Euclidean space E_5. A family of smooth lines is given in the domain Ω ⸦ E₅ so that through each point X ϵ Ω passes one line of a given family. A movable frame is chosen so that it was Frenet’s frame for the line of the given family. The integral lines of the coordinate vectors fields of this frame form a Frenet’s net. On a tangent to the line ꞷ² of this net a point  is defined in an invariant way. When the point X moves in the domain Ω the point  describes its domain . In this way we get a partial mapping  such that . The necessary and sufficient conditions for the lines belonging to 3-dimensional distributions, were quasi-double lines of the partial mapping . The subject of research is the process of partial mapping of the five-dimensional Euclidean space E₅. The purpose of the study is to find the necessary and sufficient conditions for the existence of quasi-double lines of a partial space mapping . The study used: the method of external forms of Cartan and the method of moving reper. As a result of the study, necessary and sufficient conditions for the existence of quasi-double lines for the considered partial mapping of lines belonging to three-dimensional distributions were found. The study of necessary and sufficient conditions for lines belonging to three-dimensional distributions to be quasi-double lines of a partial mapping  is considered for the first time, so the results obtained are new. The results obtained are recommended for use in the theory of differentiable mappings.