Characteristic foliations of surfaces in contact manifolds

Standard contact structure on ℝ³, version 2

Characteristic foliation on the unit sphere

Standard contact structure on ℝ³, version 3

Characteristic foliation on the unit sphere

Usual overtwisted contact structure on ℝ³

The contact structure is horizontal along the z-axis and all cylinder around this axis whose radii are multiple of π.

It is invariant under vertical translation.

A sphere whose radius lies between π and 2π

Spheres with increasing radii

You should see a video here.

Wavy disk

One sees an analogous foliation on a wavy sphere. This allows to prove that adding turns beyond the overtwisted disk does not change the contact structure.

updated on October 08 2019.