Naslov
On magnetic curves and J-trajectories in homogeneous spaces

Autori
Erjavec, Zlatko

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Conference on Geometry: Theory and Applications - Book of Abstracts / Juttler, Bert ; Lavička, Miroslav ; Sampoli M. Lucia ; Schrocker, Hans-Peter ; Žagar, Emil - Ljubljana, 2021, 9-10

Skup
Conference on Geometry: Theory and Applications (CGTA 2021)

Mjesto i datum
Gozd Martuljek, Slovenija, 20.09.2021. - 23.09.2021

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Magnetic curves ; J-trajectories ; homogeneous geometries

Sažetak
In electromagnetic theory, magnetic curves represent the trajectories of charged particles moving in Euclidean 3-space under a static magnetic field B which satisfies Gauss's law, divB=0. The magnetic field B is natural related to a two-form on R^3 and the equivalent request to Gauss's law is that the corresponding two-form is closed. Hence, the notion of a static magnetic field can be generalized to arbitrary Riemannian manifold with a closed two-form. Let (M, g, F) be a Riemannian manifold with a closed 2-form F. Then F is referred to as a magnetic field on M. By definition, a curve c(t) is called a magnetic curve if it satisfies the Lorentz equation It is known that every 3-dimensional Thurston model space admits compatible normal almost contact structure. In particular, except for the hyperbolic 3-space, every model space has a closed fundamental two-form and the compatible almost contact structure naturally induces a magnetic field. In the first part of lecture we consider magnetic curves in 3-dimensional Thurston geometries. Particularly, we discuss magnetic curves with respect to the almost contact structures of the Sol_3 space. According to Filipkiewicz there are 19 Thurston geometries in dimension four. Some of these geometries are provided with closed Kahler form which implies a magnetic field (Kahler magnetic field), but some of them are non-Kahler and theirs Kahler forms are non-closed. Hence, we can not talk about magnetic trajectories because corresponding two forms are not magnetic i.e. non-closed. However, in 4-dimensional model spaces with non-closed Kahler form (LCK spaces), it makes sense to study trajectories satisfying the equation analogous to the Lorentz equation. By definition, J-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy the equation $$\nabla_{;\dot{;\gamma};};\dot{;\gamma};=q J \dot{;\gamma};.$$ In the second part of lecture we consider magnetic curves and J-trajectories in some 4-dimensional Thurston geometries. Particularly we discuss J-trajectories in Sol_0^4.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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