Existence of a quasio-double line of a the partial mapping of the E5, genevatecl by given of smooth lines
Keywords:
Euclidean space, Frenet frame, net of Frenet, partial mapping, distribution, quasi-double lineAbstract
A family of smooth lines is given in the domain и 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.
point F2 1 is defined in an invariant way. When the point X moves in the domain Ω the point F2 1 describes its domain Ω2 1 ⊂
E5. In this way we get a partial mapping f2 1: Ω → Ω2 1 such that f2 1(X) = F2 The necessary and sufficient conditions for the lines belonging to 4-dimensional distributions, were quasidouble lines of the partial mapping f2 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 f2. 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
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