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Positive paths in diffeomorphism groups of manifolds with a contact distribution

Jakob Hedicke

TL;DR

The paper shows that positivity on the full diffeomorphism group of a cooriented contact manifold is highly flexible: any element of $\mathop{\mathrm{Diff}}_0(M)$ is connected to the identity by a null path and by a positive path, with a pivotal Chow-Rashevskii-type theorem for diffeomorphisms enabling local-to-global constructions. In $\mathbb{R}^{2n+1}$ with the standard contact structure, this yields that all diffeomorphisms are connected through positive paths, and the result extends to compactly supported diffeomorphisms on a broad class of contact manifolds using Darboux charts and fragmentation. The main method combines a local explicit construction in Darboux balls with a global assembly via the Reeb flow and the isotopy extension theorem, leading to positive isotopies between submanifolds (notably Legendrians) in thermodynamic phase spaces like $J^1\mathbb{R}^n$. Consequently, Hofer-type norms on $\mathcal{D}_c(M)$ or on Legendrian isotopy classes vanish, highlighting a stark contrast with the contactomorphism group and underscoring the non-rigidity of $\mathop{\mathrm{Diff}}(M)$ relative to $\mathop{\mathrm{Cont}}(M,\xi)$. The results have concrete implications for thermodynamic process modeling and Legendrian dynamics in phase spaces.

Abstract

Given a cooriented contact manifold $(M,ξ)$, 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 $ξ$. We show that, in contrast to the analogous notion usually considered on the group of diffeomorphisms preserving $ξ$, 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.

Positive paths in diffeomorphism groups of manifolds with a contact distribution

TL;DR

The paper shows that positivity on the full diffeomorphism group of a cooriented contact manifold is highly flexible: any element of is connected to the identity by a null path and by a positive path, with a pivotal Chow-Rashevskii-type theorem for diffeomorphisms enabling local-to-global constructions. In with the standard contact structure, this yields that all diffeomorphisms are connected through positive paths, and the result extends to compactly supported diffeomorphisms on a broad class of contact manifolds using Darboux charts and fragmentation. The main method combines a local explicit construction in Darboux balls with a global assembly via the Reeb flow and the isotopy extension theorem, leading to positive isotopies between submanifolds (notably Legendrians) in thermodynamic phase spaces like . Consequently, Hofer-type norms on or on Legendrian isotopy classes vanish, highlighting a stark contrast with the contactomorphism group and underscoring the non-rigidity of relative to . The results have concrete implications for thermodynamic process modeling and Legendrian dynamics in phase spaces.

Abstract

Given a cooriented contact manifold , it is possible to define a notion of positivity on the group of diffeomorphisms of , by looking at paths of diffeomorphisms that are positively transverse to the contact distribution . We show that, in contrast to the analogous notion usually considered on the group of diffeomorphisms preserving , positivity on is completely flexible. In particular, we show that for the standard contact structure on 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.

Paper Structure

This paper contains 15 sections, 12 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Positivity for contactomorphisms and diffeomorphisms
  3. Positive paths between submanifolds and thermodynamic processes
  4. Norms on groups of diffeomorphisms
  5. Preliminaries
  6. Notations
  7. Topologies on groups of diffeomorphisms
  8. A Chow-Rashevsky Theorem for diffeomorphisms
  9. Proofs
  10. Diffeomorphisms of Darboux-balls
  11. Proof of Theorem \ref{['thm:Rn:all']}
  12. Proof of Theorem \ref{['thm:extension']}
  13. Proof of Theorem \ref{['thm:path']}
  14. Proof of Theorem \ref{['thm:homotopic']}
  15. Proof of Theorem \ref{['cor:positive:contraction']}

Key Result

Theorem 1.1

Consider $\mathbb R^{2n+1}$ equipped with its standard contact structure. Every $f\in \mathop{\mathrm{Diff}}\limits_0(\mathbb R^{2n+1})$ is connected to $\mathrm{id}$ by a null path and a positive path of diffeomorphisms. If $f\in\mathop{\mathrm{Diff}}\limits_c(\mathbb R^{2n+1})$, it is connected to

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark 1.8
  • Corollary 1.9
  • Theorem 2.1: Agrachev09
  • ...and 5 more