EXISTENCE OF QUASI-DOUBLE LINES OF THE PAIR OF DISTRIBUTIONS

Abstract

A family of smooth lines is given in the domain Ω ⸦〖 E〗_5 so that through each point X∈Ω passes one line ω^1 of the given family. A movable frame is chosen so that it was Frenet’s frame for the line ω^1 of the given family. The integral lines of the coordinate vectors fields of this frame form a Frenet’s net. On the tangent to the line ω^3 a point  F_3^(2 ) is defined in an invariant way. When the point Х moves in the domain Ω, the point F_3^(2 ) describes its domain Ω_3^2  ⸦〖 E〗_5. In this way we get the partial mapping   f_3^2:Ω→Ω_3^2 such that f_3^2 (X)=F_3^2. It is considered the three dimensional distributions   ∆_3=(X,e ⃗_2,e ⃗_4,e ⃗_5) and   ∆_3^'=f_3^2 (∆_3 ). Definition. In the tangent to the line d⸦〖 ∆〗_3 at the point X and the tangent to the line d ̅=f_3^2 (d) at the point F_3^2 belong to the same three dimensional space (spanned by vectors e ⃗_2,e ⃗_4,e ⃗_5), then lines d and  d ̅ are called quasi-double lines of the pair (∆_3,〖∆'〗_3) of distributions in the partial mapping f_3^2. In the case when net of Frenet is cyclic net Frenet it is proved the necessary and sufficient conditions for lines d⸦〖 ∆〗_3  and d ̅=f_3^2 (d) to be quasi-double lines  of the pair of distributions (∆_3,〖∆'〗_3) in the partial mapping f_3^2