Magnetic curvature and existence of a closed magnetic geodesic on low energy levels
To a Riemannian manifold $(M,g)$ endowed with a magnetic form $\sigma $ and its Lorentz operator $\Omega $ we associate an operator $M^{\Omega }$, called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric $g$ together with terms of perturbatio...
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
November 2024
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| In: |
International mathematics research notices
Year: 2024, Volume: 2024, Issue: 21, Pages: 13586-13610 |
| ISSN: | 1687-0247 |
| DOI: | 10.1093/imrn/rnae209 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1093/imrn/rnae209
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| Author Notes: | Valerio Assenza |
| Summary: | To a Riemannian manifold $(M,g)$ endowed with a magnetic form $\sigma $ and its Lorentz operator $\Omega $ we associate an operator $M^{\Omega }$, 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 $\sigma $. From $M^{\Omega }$ we derive the magnetic sectional curvature $\textrm{Sec}^{\Omega }$ and the magnetic Ricci curvature $\textrm{Ric}^{\Omega }$ that 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 $\textrm{Ric}^{\Omega }$ 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 $\sigma $ 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 that appear when one requires $\textrm{Sec}^{\Omega }$ to be positive. |
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| Item Description: | Online veröffentlicht: 30. September 2024 Gesehen am 26.02.2025 |
| Physical Description: | Online Resource |
| ISSN: | 1687-0247 |
| DOI: | 10.1093/imrn/rnae209 |