# Finite type $\xi$-asymptotic lines of plane fields in $\mathbb{R}^3$

## TL;DR

This paper demonstrates that finite type curves can serve as $\xi$-asymptotic lines in plane fields in $
^3$, generalizing Arnold's classical results, and provides explicit examples of such lines.

## Contribution

It establishes a connection between finite type curves and $\xi$-asymptotic lines in plane fields, extending Arnold's work to a broader geometric context.

## Key findings

- Finite type curves can be realized as $\xi$-asymptotic lines in plane fields.
- Explicit example of a hyperbolic closed finite type $\xi$-asymptotic line provided.
- Results generalize Arnold's classical findings to the setting of plane fields.

## Abstract

We prove that a finite type curve is an $\xi$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $\xi$-asymptotic line. These results obtained here are generalizations, for plane fields, of the results of V. Arnold [4].