Geometric meaning of the contact condition?

Question

I am trying to understand contact structures. The definition of a contact manifold is this:

Let $M$ be a $2n + 1$-manifold and let $\omega$ be a differential $1$-form such that $\omega \wedge (d\omega)^n \neq 0$ pointwise. Then $M$ is a contact manifold and $\omega$ defines a contact structure on $M$.

Without using differential forms I think it means something like this:

A contact structure on $M$ is a smooth distribution of hyperplanes. This means that there is a smooth map $D$, the distribution, with the property that $D(m)$ is a $2n$-dimensional subspace of $T_m M$ for all $m \in M$.

At least, this is my understanding so far. Finding uninteresting examples seems easy: For example, take the sphere $S^2 \subset \mathbb R^3$ and consider tangent lines that vary smoothly with $m \in S^2$.

Writing down an explicit expression for these smooth tangent lines poses a slight challenge to me I admit and I am wondering whether if I could write down a formula that defines a tangent line to $S^2$ in $\mathbb R^3$ if it would help me gain intuitive understanding of this mysterious condition $\omega \wedge (d\omega)^n \neq 0$.

So my question is:

What are we trying to achieve by requiring $\omega \wedge (d\omega)^n \neq 0$? What does it achieve in terms of the geometric properties of the distribution of hyperplanes?

Answer 1

1) There is no line field on $S^2$. More or less this follows because $\chi(M)$ is nonzero.

2) The condition $\omega \wedge d\omega = 0$ is known as being involutive. The Frobenius theorem says that involutive distributions are integrable: there is always a chart $U$ such that, in this chart, the distribution is spanned by the first $k$ tangent vectors $\partial/\partial x_k$. A completely different-looking (but really the same) way of saying this is that, for every point $x \in M$, there is a submanifold $S \subset M$ (usually, an immersed submanifold) such that $T_x S = \mathcal D_x$ at every point $x \in S$. (I tend to think of a foliation - which is more or less equivalently an integrable distribution - as a decomposition of a manifold into immersed submanifolds.)

Then the contact condition is the exact opposite - $\omega \wedge (d\omega)^n \neq 0$ says that the distribution $\ker \omega$ is as far as possible from being integrable. (Another way of phrasing it is that $d\omega$ is a nondegenerate 2-form on $\ker \omega$.) It's very locally "twisty". (See the picture of the standard contact form on $\Bbb R^3$.)