Contact manifolds that are not cooriented

Question

One always sees the definition of a contact manifold $(X,\xi)$ as an odd dimensional manifold with a hyperplane distribution $\xi$ which can locally be expressed as $\xi = \ker \alpha$ for a 1-form $\alpha$. But in fact, it seems that in every example I know of, one always assumes that $\xi$ is cooriented, and hence we can write $\xi = \ker \alpha$ globally.

Is there a reason (other than historical) as to why coorientation wasn't built in automatically in the definition of a contact manifold? It seems strange that this isn't required in the definition.

Answer 1

As you say, the primary focus of research is on cooriented contact structures, but there is still some interest in non-coorientable contact structures (and it would become even more frustrating for those of us who are looking for information on the general case if the definition ruled out such).

For instance, David Crombecque's 2006 thesis showed that there can be some subtleties when considering the tightness of such contact structures. I quote:

In most studies, contact structures are always considered oriented. (Recall that a contact 3-manifold is always orientable but its contact structure does not have to be). It is often thought that if one has to deal with nonorientable contact structures, one may work with its orientation double cover. Although it is true that the tightness of the double cover implies the tightness of the corresponding nonorientable contact structure, our motivation is to realize that one cannot merely switch to the orientation double cover without loss of information when studying tightness. In this thesis, we systematically study the tightness of nonorientable contact structures and produce examples of 3-manifolds equipped with nonorientable tight contact structures for which the orientation double cover is overtwisted.

Answer 2

This is not an answer to your question but here is a natural example where a non cooriented contact structure arise:

the space of contact elements (the grassmanian of hyperplane) on a manifold is isomorphic $P(T^*M)=T^*M\setminus M_0/\mathbb{R}$ . The local contact forms are locally given by the Liouville form on a local section of the quotient map by the action of $\mathbb{R}$. However you cannot define a global form (you can on a double cover which corresponds to the space cooriented contact elements).

This space is of interest, for instance submanifolds of $M$ give Legendrian submanifolds of $P(T^*M)$, and isotopy give Legendrian isotopy. Section of $P(T^*M)$ which are contact submanifolds are exactly the contact structure on $M$, etc.