Abstract
Geometric calculus is a special case of the Non-Newtonian Calculus introduced by Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Also, it is a more convenient calculation method for situations where the geometric increment is more meaningful than the arithmetic increment. In this study, geometric curves are defined and geometric Frenet–Serret formulas for these curves are presented. Furthermore, we give applications of these concepts to geometric magnetic curves.
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References
Altin, A.: On the energy and pseduoangle of Frenet vector fields in \(R_{v}^{n}\). Ukr. Math. J. 63, 969–976 (2011)
Barros, M., Romero, A., Cabrerizo, J.L., Fernández, M.: The Gauss–Landau–Hall problem on Riemannian surfaces. J. Math. Phys. 46, 1 (2005)
Barros, M., Cabrerizo, J.L., Fernandez, M., Romero, A.: Magnetic vortex filament flows. J. Math. Phys. 48, 082904 (2007)
Baş, S.: A new version of spherical magnetic curves in the De-Sitter space \(S_{1}^{2}\). Symmetry 10, 606 (2018)
Boeckx, E., Vanhecke, L.: Harmonic and minimal vector fields on tangent and unit tangent bundles. Differ. Geom. Appl. 13, 77–93 (2000)
Bozkurt, Z., Gök, I., Yaylı, Y., Ekmekci, F.N.: A new approach for magnetic curves in 3D Riemannian manifolds. J. Math. Phys. 55, 053501 (2014)
Cabrerizo, J.L.: Magnetic fields in 2D and 3D sphere. J. Nonlinear Math. Phys. 20, 440 (2013)
Cabrerizo, J.L., Fernandez, M., Gomez, J.S.: On the existence of almost contact structure and the contact magnetic field. Acta Math. Hung. 125, 191–199 (2009)
Chacon, P.M., Naveira, A.M., Osaka, J.: Corrected energy of distributions on Riemannian Manifolds. Osaka J. Math. 41, 97–105 (2004)
Druta-Romaniuc, S.L., Munteanu, M.I.: Magnetic curves corresponding to killing magnetic fields in \(\cal{E} ^{3}\). J. Math. Phys. 52, 113506:1 (2011)
Druţă-Romaniuc, S.L., Munteanu, M.I.: Killing magnetic curves in a Minkowski 3-space. Nonlinear Anal. Real World Appl. 14, 383–396 (2013)
Ekinci, A., Baş, S., Körpınar, T., Körpınar, Z.: Optical new geometric magnetic curves according to geometric Frenet formulas (in review) (2023)
Gil-Medrano, O.: Relationship between volume and energy of vector field. Differ. Geom. Appl. 15, 137–152 (2001)
Gluck, H., Ziller, W.: On the volume of the unit vector fields on the three sphere. Comment Math. Helv. 6, 177 (1986)
Gonzalez-Davila, J.C., Vanhecke, L.: Examples of minimal unit vector fields. Ann. Glob. Anal. Geom. 18, 385 (2000)
Grossman, M., Katz, R.: Non-Newtonian Calculus. Lee Press, Pigeon Cove (1972)
Johnson, D.L.: Volume of flows. Proc. Am. Math. Soc. 104, 923–931 (1988)
Körpınar, T.: New characterization for minimizing energy of biharmonic particles in Heisenberg spacetime. Int. J. Phys. 53, 3208–3218 (2014)
Körpınar, T.: On \(\textbf{T}\)-magnetic biharmonic particles with energy and angle in the three dimensional Heisenberg group \(\mathbb{H}\). Adv. Appl. Clifford Algebras 28, 9 (2018a)
Körpınar, T.: A new version of normal magnetic force particles in 3D Heisenberg space. Adv. Appl. Clifford Algebras 28, 83 (2018b)
Körpınar, T., Demirkol, R.C.: Frictional magnetic curves in 3D Riemannian manifolds. Int. J. Geom. Methods Mod. Phys. 15, 1850020–1 (2018)
Körpınar, T., Demirkol, R.C.: Electromagnetic curves of the linearly polarized light wave along anoptical fiber in a 3D semi-Riemannian manifold. J. Mod. Opt. 66, 857–867 (2019)
Körpınar, T., Demirkol, R.C.: Electromagnetic curves of the linearly polarized light wave along anoptical fiber in a 3D Riemannian manifold with Bishop equations. Optik 200, 163334 (2020a)
Körpınar, T., Demirkol, R.C.: Electromagnetic curves of the polarized light wave along the optical fiber in De-Sitter 2-space \(S_{1}^{2}\). Indian J. Phys. 95, 147–156 (2020b)
Körpinar, T., Demirkol, R.C., Körpınar, Z.: Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in Minkowski space with Bishop equations. Eur. Phys. J. D 73, 1–11 (2019)
Körpinar, T., Demirkol, R.C., Khalil, E.M., Korpinar, Z., Baleanu, D.: Quasi binormal Schrodinger evolution of wave polarizatıon field of light wıth repulsive type. Phys. Scr. 96(4), 045104 (2021)
Özdemir, Z.: A new calculus for the treatment of Rytov’s law in the optical fiber. Optik 216, 164892 (2020)
Sunada, T.: Magnetic flows on a Riemann surface. In: Proceedings of KAIST Mathematics workshop, p. 93 (1993)
Wiegmink, G.: Total bending of vector fields on Riemannian manifolds. Math. Ann. 303, 325–344 (1995)
Wood, C.M.: On the energy of a unit vector field. Geom. Dedicata 64, 319–330 (1997)
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Ekinci, A., Bas, S., Körpinar, T. et al. New geometric magnetic energy according to geometric Frenet formulas. Opt Quant Electron 56, 81 (2024). https://doi.org/10.1007/s11082-023-05569-z
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DOI: https://doi.org/10.1007/s11082-023-05569-z