NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF QUASI-DOUBLE LINES OF A PAIR OF DISTRIBUTIONS IN Ε_5

Abstract

In the domain Ω ⊂ E₅ consider a set of smooth lines such that through each point X ∈ Ω there passes exactly one line ω¹ from this set. The moving frame ℜ is a Frenet frame [1] associated with the line ω¹ of the given set. The integral lines of the vector fields e _i form the Frenet net Σ₅ [1]. There exists a point   on the tangent of the line ω⁵ of the Frenet net. When the point X moves within the domain Ω, the point   describes out its domain  in . The partial mapping is defined  such that  such that . Pairs of 3-dimensional distributions are considered:   different) in the .In the case when the Frenet net is a cyclic Frenet net, lines belonging to 3-dimensional distributions are considered. Necessary and sufficient conditions are proved for these lines to be quasi-double lines of a pair of 3-dimensional distributions in the parietal mapping .