Canonical frames in contact 3-manifolds and applications
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arXiv
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| Format: | Preprint |
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2026
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| author | Ferreira, Brayan Miranda, Marcelo Vicente, Alejandro |
| author_facet | Ferreira, Brayan Miranda, Marcelo Vicente, Alejandro |
| contents | We study contact 3-manifolds $Y$ with a special global frame inspired by Cartan's structure equations. This frame is dual to a generalized Finsler structure defined by Bryant. We present some examples and rigidity results on the class of manifolds whose frame satisfies certain natural conditions on a scalar function $K\colon Y\to \mathbb{R}$, related to the frame. This function realizes the curvature when $Y$ is the unit tangent bundle with respect to a metric on a surface. As applications, we obtain sharp estimates for the action of a Reeb orbit in terms of this scalar function, under the assumption that the frame satisfies specific conditions. In particular, we recover a classical upper bound on the systole of positively curved metrics on $S^2$ due to Toponogov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_30027 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Canonical frames in contact 3-manifolds and applications Ferreira, Brayan Miranda, Marcelo Vicente, Alejandro Symplectic Geometry Differential Geometry 53D25, 53D10 We study contact 3-manifolds $Y$ with a special global frame inspired by Cartan's structure equations. This frame is dual to a generalized Finsler structure defined by Bryant. We present some examples and rigidity results on the class of manifolds whose frame satisfies certain natural conditions on a scalar function $K\colon Y\to \mathbb{R}$, related to the frame. This function realizes the curvature when $Y$ is the unit tangent bundle with respect to a metric on a surface. As applications, we obtain sharp estimates for the action of a Reeb orbit in terms of this scalar function, under the assumption that the frame satisfies specific conditions. In particular, we recover a classical upper bound on the systole of positively curved metrics on $S^2$ due to Toponogov. |
| title | Canonical frames in contact 3-manifolds and applications |
| topic | Symplectic Geometry Differential Geometry 53D25, 53D10 |
| url | https://arxiv.org/abs/2603.30027 |