Contact of a regular surface in Euclidean 3-space with cylinders and cubic binary differential equations

Abstract

We investigate the contact types of a regular surface in the Euclidean 3-space ℝ³ with right circular cylinders. We present the conditions for existence of cylinders with A₁, A₂, A₃, A₄, A₅, D₄, and D₅ contacts with a given surface. We also investigate the kernel field of A_≥3-contact cylinders on the surface. This is defined by a cubic binary differential equation and we classify singularity types of its flow in the generic context.