Positive paths in diffeomorphism groups of manifolds with a contact distribution

TL;DR

This paper explores the concept of positive paths in the diffeomorphism groups of contact manifolds, revealing their flexibility and applications to Legendrian submanifolds and thermodynamics.

Contribution

It demonstrates that positivity in the full diffeomorphism group of contact manifolds is highly flexible, contrasting with the restricted case of contact-preserving diffeomorphisms, and applies this to Legendrian and thermodynamic problems.

Findings

Any two diffeomorphisms of the standard contact Euclidean space are connected by a positive path.

Positivity on the full diffeomorphism group is completely flexible for a broad class of contact manifolds.

The results answer a question about Legendrians in thermodynamic phase space.

Abstract

Given a cooriented contact manifold $(M,\xi)$, it is possible to define a notion of positivity on the group $\mathrm{Diff}(M)$ of diffeomorphisms of $M$, by looking at paths of diffeomorphisms that are positively transverse to the contact distribution $\xi$. We show that, in contrast to the analogous notion usually considered on the group of diffeomorphisms preserving $\xi$, positivity on $\mathrm{Diff}(M)$ is completely flexible. In particular, we show that for the standard contact structure on $\mathbb{R}^{2n+1}$ any two diffeomorphisms are connected by a positive path. This result generalizes to compactly supported diffeomorphisms on a large class of contact manifolds. As an application we answer a question about Legendrians in thermodynamic phase space posed by Entov, Polterovich and Ryzhik in the context of thermodynamic processes.