Contact structures and Beltrami fields on the torus and the sphere
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912041495691264 |
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| author | Peralta-Salas, Daniel Slobodeanu, Radu |
| author_facet | Peralta-Salas, Daniel Slobodeanu, Radu |
| contents | We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields using suitable families of Jacobi or trigonometric polynomials. As a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk and Massot (2012) does not hold for weakly compatible metric as it was conjectured. We also establish a geometric rigidity for tight contact structures by showing that any contact form on the 3-sphere admitting a compatible metric that is the round one is isometric, up to a constant factor, to the standard (tight) contact form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_10185 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Contact structures and Beltrami fields on the torus and the sphere Peralta-Salas, Daniel Slobodeanu, Radu Differential Geometry Mathematical Physics Dynamical Systems Symplectic Geometry We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields using suitable families of Jacobi or trigonometric polynomials. As a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk and Massot (2012) does not hold for weakly compatible metric as it was conjectured. We also establish a geometric rigidity for tight contact structures by showing that any contact form on the 3-sphere admitting a compatible metric that is the round one is isometric, up to a constant factor, to the standard (tight) contact form. |
| title | Contact structures and Beltrami fields on the torus and the sphere |
| topic | Differential Geometry Mathematical Physics Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2004.10185 |