1. Magnetic Curvature and Existence of a Closed Magnetic Geodesic on Low Energy Levels.
- Author
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Assenza, Valerio
- Subjects
- *
ENERGY levels (Quantum mechanics) , *RIEMANNIAN metric , *ORBITS (Astronomy) , *CURVATURE , *GEODESICS - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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