The regularity and products in contact geometry
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909443949592576 |
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| author | Grabowska, Katarzyna Grabowski, Janusz |
| author_facet | Grabowska, Katarzyna Grabowski, Janusz |
| contents | We study regular contact manifolds $(M,η)$ whose Reeb vector field is complete and prove that they are canonically principal bundles with the structure group $S^1$ or $\mathbb{R}$. For compact $M$, our proof is very short and elementary and covers the celebrated Boothby-Wang theorem, but we do not assume compactness from the very beginning. However, to prove our result in full generality we use some topological tools adapted to smooth fibrations. In the second part of the paper, we describe a natural concept of contact products of general contact manifolds as well as a product of principal contact manifolds, which exists if the periods of the Reeb vector fields are commensurate, and corresponds to the construction of products of prequantization bundles of symplectic manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20491 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The regularity and products in contact geometry Grabowska, Katarzyna Grabowski, Janusz Symplectic Geometry Mathematical Physics Differential Geometry 53D10, 53D35, 37C10, 37C86 We study regular contact manifolds $(M,η)$ whose Reeb vector field is complete and prove that they are canonically principal bundles with the structure group $S^1$ or $\mathbb{R}$. For compact $M$, our proof is very short and elementary and covers the celebrated Boothby-Wang theorem, but we do not assume compactness from the very beginning. However, to prove our result in full generality we use some topological tools adapted to smooth fibrations. In the second part of the paper, we describe a natural concept of contact products of general contact manifolds as well as a product of principal contact manifolds, which exists if the periods of the Reeb vector fields are commensurate, and corresponds to the construction of products of prequantization bundles of symplectic manifolds. |
| title | The regularity and products in contact geometry |
| topic | Symplectic Geometry Mathematical Physics Differential Geometry 53D10, 53D35, 37C10, 37C86 |
| url | https://arxiv.org/abs/2412.20491 |