NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF A QUASIO-DOUBLE LINE OF A PAIR IN EUCLIDEAN SPACE

Abstract

This research belongs to the rapidly developing areas of modern differential geometry, namely: the geometry of differentiable mappings of smooth manifolds and the geometry of a network on smooth manifolds. The research examines the partial mappings of the Euclidean space generated by a given distribution, and reveals the close connections between the theories of mappings, networks and distributions. Specifying a 6-dimensional distribution in some areas of Euclidean space is determined in an invariant way, orthogonally complementary to the given distribution. The purpose of the study is to prove the necessary and sufficient conditions for the existence of a quasi-double line. Research methods: Cartan's method of external forms and a movable benchmark.This study relates to the rapidly developing areas of modern differential geometry, which is: the geometry of differentiable maps of smooth manifolds and network geometry on smooth manifolds. It is considered a set of smooth lines such that through a point passed one line of given set in domain . The moving frame is frame of Frenet for the line of the given set. Integral lines of the vector fields are formed net of Frenet. There exists a point on the tangent of the line . When a point is shifted in the domain , the point describes it’s domain in . It is defined the partial mapping , such that . Necessary and sufficient conditions of existence of a quasio-double line of a pair are proved.