A new metric on the contactomorphism group of orderable contact manifolds
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908297420865536 |
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| author | Nakamura, Lukas |
| author_facet | Nakamura, Lukas |
| contents | We introduce a pseudo-metric on the contactomorphism group of any contact manifold $(M,ξ)$ with a cooriented contact structure $ξ$. It is the contact analogue of a corresponding semi-norm in Hofer's geometry, and on certain classes of contact manifolds, its lift to the universal cover can be viewed as a continuous version of the integer valued bi-invariant metric introduced by Fraser, Polterovich, and Rosen. We show that it is non-degenerate if and only if $(M,ξ)$ is strongly orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. In particular, the interval topology is Hausdorff whenever it is non-trivial, which answers a question of Chernov and Nemirovski. We discuss analogous results for isotopy classes of Legendrians and universal covers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10905 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new metric on the contactomorphism group of orderable contact manifolds Nakamura, Lukas Symplectic Geometry 53D10, 53D35 We introduce a pseudo-metric on the contactomorphism group of any contact manifold $(M,ξ)$ with a cooriented contact structure $ξ$. It is the contact analogue of a corresponding semi-norm in Hofer's geometry, and on certain classes of contact manifolds, its lift to the universal cover can be viewed as a continuous version of the integer valued bi-invariant metric introduced by Fraser, Polterovich, and Rosen. We show that it is non-degenerate if and only if $(M,ξ)$ is strongly orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. In particular, the interval topology is Hausdorff whenever it is non-trivial, which answers a question of Chernov and Nemirovski. We discuss analogous results for isotopy classes of Legendrians and universal covers. |
| title | A new metric on the contactomorphism group of orderable contact manifolds |
| topic | Symplectic Geometry 53D10, 53D35 |
| url | https://arxiv.org/abs/2307.10905 |