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 $\mathbb{R}^3$ with right circular cylinders. We present the conditions for existence of cylinders with $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $D_4$, and $D_5$ contacts with a given surface. We also investigate the kernel field of $A_{\ge 3}$-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.