About One Partial Mapping of Euclidean Space E5

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 e¹, e², ^e3, e⁴, e⁵ form the Frenet net Σ₅ [1]. There exists a point F₃ ² on the tangent to the line ω₃ of the Frenet net. When the point X moves within the domain Ω, the point F₃ ² describes its domain Ω₃ ² in E₅. The partial mapping is defined as f₃ ²: Ω → Ω₃ ² such that f₃ ²(X) = F₃ ².
We consider the four-dimensional distributions Δ4 = (X, e₁, e₂, e₄, e₅) and Δ^′₄ = f₃ ²(Δ₄). In the case where the Frenet net is cyclic, the necessary and sufficient conditions for the existence of quasi-double lines of the pair (Δ₄, Δ′₄) of four-dimensional distributions under the partial mapping f₃ ² in the space E₅ are established.