Killing vectors and magnetic curves associated to Bott connection in Heisenberg group. (English) Zbl 07795054
Summary: In this paper, we define the notion of Bott connection in the Heisenberg group \((\mathbb{H}_3,g)\) and derive the expression of the Bott connection by using the Levi-Civita connection. Moreover, we derive the expressions of killing vector fields by using the killing equation and obtain some explicit formulas for killing magnetic curves associated to the Bott connection. Furthermore, we give some examples of killing magnetic curves.
References:
| [1] | Özgür, C., On magnetic curves in \(3\)-dimensional Heisenberg group, Proc. Inst. Math. Mech.43 (2017) 278-286. · Zbl 1395.53055 |
| [2] | Özdemi̇r, Z., Gök, İ., Yayli, Y. and Ekmekci̇, F. N., Notes on magnetic curves in \(3D\) semi Riemannian manifolds, Turk. J. Math.39 (2015) 412-426. · Zbl 1345.53025 |
| [3] | Drută-Romaniuc, S. L., Inoguchi, J., Munteanu, M. I. and Nistor, A. I., Magnetic curves in cosymplectic manifolds, Rep. Math. Phys.78 (2016) 33-48. · Zbl 1353.53033 |
| [4] | Drută-Romaniuc, S. L., Inoguchi, J., Munteanu, M. I. and Nistor, A. I., Magnetic curves in Sasakian manifolds, J. Nonlinear. Math. Phys.22 (2015) 428-447. · Zbl 1420.53035 |
| [5] | Drută-Romaniuc, S. L. and Munteanu, M. I., Magnetic curves corresponding to killing magnetic fields in \(\mathbb{E}^3\), J. Math. Phys.52 (2011) 113506. · Zbl 1272.37020 |
| [6] | Drută-Romaniuc, S. L. and Munteanu, M. I., Killing magnetic curves in a Minkowski 3-space, Nonlinear Anal.-Real.14 (2013) 383-396. · Zbl 1255.53016 |
| [7] | Munteanua, M. I. and Nistor, A., The classification of killing magnetic curves in \(\mathbb{S}^2\times\mathbb{R} \), J. Geom. Phys.62 (2012) 170-182. · Zbl 1239.78003 |
| [8] | Erjavec, Z., On killing magnetic curves in SL(2, R) geometry, Rep. Math. Phys.84 (2019) 333-350. · Zbl 1441.78006 |
| [9] | Erjavec, Z. and Inoguchi, J., Killing Magnetic Curves in Sol space, Math. Phys. Anal. Geom.21 (2018) 15. · Zbl 1395.53084 |
| [10] | Iqbal, Z., Sengupta, J. and Chakraborty, S., Magnetic trajectories corresponding to killing magnetic fields in a three-dimensional warped product, Int. J. Geom. Methods Mod. Phys.17 (2020) 2050212. · Zbl 07819422 |
| [11] | Calvaruso, G., Munteanu, M. I. and Perrone, A., Killing magnetic curves in three-dimensional almost paracontact manifolds, J. Math. Anal. Appl.426 (2015) 423-439. · Zbl 1322.53033 |
| [12] | Erjavec, Z. and Inoguchi, J., On Magnetic Curves in Almost Cosymplectic Sol Space, Results Math.75 (2020) 113. · Zbl 1442.53021 |
| [13] | Bejan, C. L. and Drută-Romaniuc, S. L., Walker manifolds and killing magnetic curves, Differ. Geom. Appl.35 (2014) 106-116. · Zbl 1322.53074 |
| [14] | Jiang, X. and Sun, J., Local geometric properties of the lightlike killing magnetic curves in de Sitter 3-space, Aims. Math.6 (2021) 12543-12559. · Zbl 1510.53049 |
| [15] | Sun, J., Singularity properties of killing magnetic curves in Minkowski 3-space, Int. J. Geom. Methods Mod. Phys.16 (2019) 1950123. · Zbl 1421.78015 |
| [16] | Sun, J., Singularity properties of null killing magnetic curves in Minkowski 3-space, Int. J. Geom. Methods. Mod. Phys.17 (2020) 2050141. · Zbl 07812079 |
| [17] | Wu, T. and Wang, Y., Affine Ricci solitons associated to the Bott connection on three-dimensional Lorentzian Lie group, Turk. J. Math.45 (2021) 2773-2816. · Zbl 1508.53027 |
| [18] | Wu, T. and Wang, Y., Codazzi Tensors and the Quasi-Statistical structure Associated with affine connections on Three-Dimensional Lorentzian Lie Groups, Symmetry13 (2021) 1459. |
| [19] | Aikou, T., Applications of Bott connection to Finsler geometry, in Proc. Colloquium Steps in Differential Geometry (Hungary, , 2000), pp. 25-30. · Zbl 0997.53055 |
| [20] | Z. Giunashvili, Bott Connection and Generalized Functions on Poisson manifold (2003), arXiv preprint math/0301364. |
| [21] | Li, Y. and Tuncer, O. O., On (contra) pedals and (anti) orthotomics of frontals in de Sitter 2-space, Math. Meth. Appl. Sci.1 (2023) 1-15. |
| [22] | Li, Y., Erdoğdu, M. and Yavuz, A., Differential geometric approach of Betchow-Da Rios Soliton Equation, Hacet. J. Math. Stat.1 (2022) 1-12. |
| [23] | Li, Y., Abolarinwa, A., Alkhaldi, A. H. and Ali, A., Some inequalities of Hardy type related to Witten-Laplace operator on smooth metric measure spaces, Mathematics.10 (2022) 4580. |
| [24] | Li, Y., Aldossary, M. T. and Abdel-Baky, R. A., Spacelike circular surfaces in Minkowski 3-space, Symmetry.15 (2023) 173. |
| [25] | Li, Y., Chen, Z., Nazra, S. H. and Abdel-Baky, R. A., Singularities for timelike developable surfaces in Minkowski 3-space, Symmetry.15 (2023) 277. |
| [26] | Li, Y., Alkhaldi, A. H., Ali, A., Abdel-Baky, R. A. and Saad, M. K., Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean 3-space, Aims Math.8 (2023) 13875-13888. |
| [27] | Qian, Y. and Yu, D., Rates of approximation by neural network interpolation operators, Appl. Math. Comput.41 (2022) 126781. · Zbl 1510.41005 |
| [28] | Li, Y., Eren, K., Ayvacı, K. H. and Ersoy, S., The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, Aims Math.8 (2023) 2226-2239. |
| [29] | Li, Y., Abdel-Salam, A. A. and Saad, M. K., Primitivoids of curves in Minkowski plane, Aims Math.8 (2023) 2386-2406. |
| [30] | Li, W. and Liu, H., Gauss-Bonnet theorem in the universal covering group of Euclidean motion group \(E(2)\) with the general left-invariant metric, J. Nonlinear Math. Phy.29 (2022) 626-657. · Zbl 1497.53071 |
| [31] | Liu, H. and Guan, J., Sub-Lorentzian geometry of curves and surfaces in a Lorentzian Lie group, Adv. Math. Phys. (2022). · Zbl 1500.53040 |
| [32] | Liu, H. and Guan, J., Gauss-Bonnet theorems for Lorentzian and spacelike surfaces associated to canonical connections in the Lorentzian Heisenberg group, Sib. Math. J.64 (2023) 471-499. · Zbl 1519.53057 |
| [33] | Derdzinski, A. and Shen, C., Codazzi tensor fields, Curvature and Pontryagin forms, P. Lond. Math. Soc.47 (1983) 15-26. · Zbl 0519.53015 |
| [34] | López, J. A. A. and Tondeur, P., Hodge decomposition along the leaves of a Riemannian foliation, J. Funct. Anal.99 (1991) 443-458. · Zbl 0746.58011 |
| [35] | Baudoin, F., Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations, J. Eur. Math. Soc.1 (2016) 259-321. · Zbl 1379.53035 |
| [36] | R. K. Hladky, Connections and curvature in sub-Riemannian geometry, (2009), arXiv preprint arXiv:0912.3535. · Zbl 1264.53040 |
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