Magnetic curvature and existence of a closed magnetic geodesic on low energy levels
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912018387173376 |
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| author | Assenza, Valerio |
| author_facet | Assenza, Valerio |
| contents | To a Riemannian manifold $(M, g)$ endowed with a magnetic form $σ$ and its Lorentz operator $Ω$ we associate an operator $M^Ω$, called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric $g$ together with terms of perturbation due to the magnetic interaction of $σ$. From $M^Ω$ we derive the magnetic sectional curvature $Sec^Ω$ and the magnetic Ricci curvature $Ric^Ω$ which generalize in arbitrary dimension the already known notion of magnetic curvature previously considered by several authors on surfaces. On closed manifolds, under the assumption of $Ric^Ω$ being positive on an energy level below the Mañé critical value, with a Bonnet-Myers argument, we establish the existence of a contractible periodic orbit. In particular, when $σ$ is nowhere vanishing, this implies the existence of a contractible periodic orbit on every energy level close to zero. Finally, on closed oriented even dimensional manifolds, we discuss about the topological restrictions which appear when one requires $Sec^Ω$ to be positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03159 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Magnetic curvature and existence of a closed magnetic geodesic on low energy levels Assenza, Valerio Symplectic Geometry Differential Geometry Dynamical Systems 53D25, 53C21 (primary) 37N05 (secondary) To a Riemannian manifold $(M, g)$ endowed with a magnetic form $σ$ and its Lorentz operator $Ω$ we associate an operator $M^Ω$, called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric $g$ together with terms of perturbation due to the magnetic interaction of $σ$. From $M^Ω$ we derive the magnetic sectional curvature $Sec^Ω$ and the magnetic Ricci curvature $Ric^Ω$ which generalize in arbitrary dimension the already known notion of magnetic curvature previously considered by several authors on surfaces. On closed manifolds, under the assumption of $Ric^Ω$ being positive on an energy level below the Mañé critical value, with a Bonnet-Myers argument, we establish the existence of a contractible periodic orbit. In particular, when $σ$ is nowhere vanishing, this implies the existence of a contractible periodic orbit on every energy level close to zero. Finally, on closed oriented even dimensional manifolds, we discuss about the topological restrictions which appear when one requires $Sec^Ω$ to be positive. |
| title | Magnetic curvature and existence of a closed magnetic geodesic on low energy levels |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems 53D25, 53C21 (primary) 37N05 (secondary) |
| url | https://arxiv.org/abs/2309.03159 |