Asymptotic Nets in the Improper Three-dimensional Affine Hyperplane

Abstract

The two – dimensional surface  is considered in the four-dimensional extended affine space   Α₄=Α₄∪Α₃^*, where  Α₃^* is a three-dimensional improper hyperplane carrying the structure of the projective space P₃. A moving frame R = (A,A_i,A_α )(i=1,2; α=3,4) is attached to the surface   according to a certain rule, where A ∈ V₂, {A_i,A_α } ⊂ Α₃*. We choose a conjugated net (ω¹, ω² ) arranged by geodesic lines on the surface V₂   so that the straigt lines (AA_i) represent tangent to the lines ωⁱ. When the point   moves on the surface  V₂  then each of the points A_i, A_α describes two-dimensional surfaces  (A_i ),(A_α ) in the hyperplane A₃*, on which naturally arise nets of lines   (ω¹, ω² ). The cases when the nets are asymptotic are considered.