ABOUT EXISTENCE OF A QUASIO-DOUBLE LINES OF THE PARTIAL MAPPING IN SPACE

Abstract

It is considered the problem related to partial mapping of 6- dimensional Euclidean space  

 A family of smooth lines is given in the domain  so that through each point  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  of this net a point  is defined in an invariant way.

When the point  moves in the domain  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 4-dimensional distributions, were quasi-double lines of the partial mapping .

The subject of research is the process of partial mapping of the six-dimensional Euclidean space . 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 4-dimensional distributions were found.

The study of necessary and sufficient conditions for lines belonging to 4-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.