2024/04/15 by Jorge Luiz Deolindo Silva, Deolindo-Silva, Jorge Luiz
Mathematics · #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2404.09963
openalex publication_date 2024/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A smooth ruled surface in 4-space has only parabolic points or inflection points of real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along which the parallel projection exhibits \mathcal A-singularities of type butterfly or worse. In particular, such parabolic point can be classified as butterfly hyperbolic, parabolic, or elliptic point depending on the value of the discriminant of a binary differential equation (BDE). Also, whenever such discriminant is positive, we ensure that the integral curves of these directions form a pair of foliations on the ruled surface. Moreover, the set of points that nullify the discriminant is a regular curve transverse to the regular curve formed by inflection points of real type. Finally, using a particular projective transformation, we obtain a simple parametrization of the ruled surface such that the moduli of its 5-jet identify a butterfly hyperbolic/parabolic/elliptic point, as well as we get the stable configurations of the solutions of BDE in the discriminant curve.