Positive paths in diffeomorphism groups of manifolds with a contact distribution
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909961581232128 |
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| author | Hedicke, Jakob |
| author_facet | Hedicke, Jakob |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07279 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive paths in diffeomorphism groups of manifolds with a contact distribution Hedicke, Jakob Symplectic Geometry Mathematical Physics Differential Geometry 53D10, 57S05 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. |
| title | Positive paths in diffeomorphism groups of manifolds with a contact distribution |
| topic | Symplectic Geometry Mathematical Physics Differential Geometry 53D10, 57S05 |
| url | https://arxiv.org/abs/2507.07279 |