2018/11/18 by Marcos Craizer, Craizer, Marcos, Ronaldo Garcia +1
Engineering · Mathematics · #53A05 #53A15 #Advanced Numerical Analysis Techniques #Analytic Number Theory Research #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1811.07331
openalex publication_date 2018/11/18 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
There is a natural duality between line congruences in ℝ3 and surfaces in ℝ4 that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat ridge curves, while the subparabolic curves of a line congruence are taken to certain curves on the surface that we call flat subparabolic curves. In this paper, we discuss these relations and describe the generic behavior of the subparabolic curves at the discriminant curve of the line congruence, or equivalently, the parabolic curve of the dual surface. We also discuss Loewner's conjectures under the duality viewpoint.