Magnetic trajectories on tangent sphere bundle with g-natural metrics
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Magnetic trajectories on tangent sphere bundle with g‑natural metrics Mohamed Tahar Kadaoui Abbassi1 · Noura Amri1 · Marian Ioan Munteanu2 Received: 1 September 2019 / Accepted: 31 July 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study magnetic trajectories in the unit tangent sphere bundle with pseudo-Riemannian g-natural metrics of a Riemannian manifold. A high interest is dedicated to the case when the base manifold is a space form and when the metric is of Kaluza–Klein type. Slant curves are obtained when a certain conservation law is satisfied. We give a complete classification of slant magnetic curves (respectively, geodesics) on T1 M , when M is a twodimensional Riemannian manifold of constant curvature. Keywords Magnetic curves · Geodesics · Unit tangent (sphere)bundle · g-natural metrics Mathematics Subject Classification 53B21 · 53C15 · 53C25 · 53C80
1 Introduction About 45 years ago, a very interesting result concerning geodesics on the unit tangent bundle of the 2-sphere was proved in [9] by Klingenberg and Sasaki. More precisely, they showed that any geodesic on T1 𝕊2 , endowed with the restriction of the Sasaki metric from T𝕊2 , is a circle x(s) on 𝕊2 together with a unitary vector field V(s) along x, making constant angle with x(s). See also [12] for a classification of geodesics on the unit tangent bundles over space forms of arbitrary dimension. These results inspired the authors of [8] to consider magnetic curves in T1 M , since it admits a natural contact metric structure and therefore, contact magnetic fields on T1 M may be considered. In the same paper [8], magnetic equations on T1 M were obtained * Marian. Ioan Munteanu [email protected] Mohamed Tahar Kadaoui Abbassi [email protected] Noura Amri [email protected] 1
Département des Mathématiques, Faculté des sciences Dhar El Mahraz, Université Sidi Mohamed Ben Abdallah, B.P. 1796, Fès‑Atlas, Fès, Morocco
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Department of Mathematics, Alexandru Ioan Cuza University of Iasi, Bd. Carol I, No. 11, 700506 Iasi, Romania
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for an arbitrary Riemannian manifold M, in particular when M is a space form of constant sectional curvature c. It was proved that if c = 1 , then every contact unit speed magnetic curve is slant. On contrary, if c ≠ 1 , this property is not automatically satisfied and it is equivalent to a certain conservation law. For more details on the geometry of these curves see [8] and [7]. In [1], the first author and G. Calvaruso constructed a three-parameter family of contact metric structures on the unit tangent bundle T1 M of a Riemannian manifold M, for which the associated metrics are g-natural metrics. In the present paper we endow the unit tangent bundle with an element of this family, i.e. with a pseudo-Riemannian g-natural metric and a com̃ 𝜂, ̃ , and we investigate magnetic trajectories. ̃ 𝜓, ̃ 𝜉) patible almost contact metric structure (G, As the structure is
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