1 Introduction

A contact structure on a \(3\)-manifold \(M\) is a maximally non-integrable rank \(2\) subbundle \(\xi\) of the tangent bundle of \(M\). If \(\alpha\) is a \(1\)-form on \(M\) whose kernel is \(\xi\), then \(\xi\) is a contact structure if and only if \(\alpha\wedge\mathrm{d}\alpha\neq 0\). A curve \(\eta\) in a contact \(3\)-manifold \((M,\xi)\) is called Legendrian, whenever \(\eta^*\alpha=0\) for some (local) \(1\)-form \(\alpha\) defining \(\xi\).

The purpose of this note is to give a detailed proof of the following statement which is often used in contact geometry and Legendrian knot theory.

Whereas this theorem is a special case of Gromov’s \(h\)-principle for Legendrian immersions [4], the curve-case can be treated by more elementary techniques. Sketches of proofs of Theorem 1.1 have already appeared in the literature, see for example [1], [2] or [3]. Exploiting the fact that every contact \(3\)-manifold is locally contactomorphic to \(\mathbb R^3\) equipped with the standard contact structure defined by \(\alpha=\mathrm{d}z -y \mathrm{d}x\), Etnyre and Geiges indicate that either the front-projection \((x,z)\) of a given curve \((x,y,z)\) can be approximated by a zig-zag-curve whose slope approximates the \(y\)-component of the curve or the Lagrangian projection \((x,y)\) can be approximated by a curve whose area integral approximates the \(z\) component of the curve, which can be achieved by adding small negatively or positively oriented loops.

Here, we give a different and analytically rigorous proof of Theorem 1.1 by using convex integration. Our proof has the advantage of providing a constructive approximation. In particular, in the case of a continuous curve in \(\mathbb R^3\) equipped with the standard contact structure, we obtain an explicit Legendrian curve given in terms of an elementary integral. For instance, we obtain an explicit solution to the “parallel parking problem” in Example 3.1. Example 3.2 shows how our technique recovers the zig-zag-curves and the small loops in the front - respectively Lagrangian projections.