Czechoslovak Mathematical Journal, Vol. 71, No. 1, pp. 75-109, 2021
On $g$-natural conformal vector fields on unit tangent bundles
Mohamed Tahar Kadaoui Abbassi, Noura Amri
Received April 24, 2019. Published online August 17, 2020.
Abstract: We study conformal and Killing vector fields on the unit tangent bundle, over a Riemannian manifold, equipped with an arbitrary pseudo-Riemannian $g$-natural metric. We characterize the conformal and Killing conditions for classical lifts of vector fields and we give a full classification of conformal fiber-preserving vector fields on the unit tangent bundle endowed with an arbitrary pseudo-Riemannian Kaluza-Klein type metric.
Keywords: conformal vector field; unit tangent bundle; $g$-natural metric
References: [1] M. T. K. Abbassi: Métriques naturelles sur le fibré tangent à une variété Riemannienne. Editions universitaires europeennes, Paris (2012). (In French.)
[2] M. T. K. Abbassi, N. Amri, G. Calvaruso: Kaluza-Klein type Ricci solitons on unit tangent sphere bundles. Differ. Geom. Appl. 59 (2018), 184-203. DOI 10.1016/j.difgeo.2018.04.006 | MR 3804828 | Zbl 1391.53053
[3] M. T. K. Abbassi, G. Calvaruso: $g$-natural contact metrics on unit tangent sphere bundles. Monaths. Math. 151 (2007), 89-109. DOI 10.1007/s00605-006-0421-9 | MR 2322938 | Zbl 1128.53049
[4] M. T. K. Abbassi, G. Calvaruso: $g$-natural metrics of constant curvature on unit tangent sphere bundles. Arch. Math., Brno 48 (2012), 81-95. DOI 10.5817/AM2012-2-81 | MR 2946208 | Zbl 1274.53112
[5] M. T. K. Abbassi, O. Kowalski: On $g$-natural metrics with constant scalar curvature on unit tangent sphere bundles. Topics in Almost Hermitian Geometry and Related Fields. World Scientific, Hackensack (2005), 1-29. DOI 10.1142/9789812701701_0001 | MR 2181488 | Zbl 1107.53023
[6] M. T. K. Abbassi, O. Kowalski: Naturality of homogeneous metrics on Stiefel manifolds $SO(m+1)/SO(m-1)$. Differ. Geom. Appl. 28 (2010), 131-139. DOI 10.1016/j.difgeo.2009.05.007 | MR 2594457 | Zbl 1190.53020
[7] M. T. K. Abbassi, O. Kowalski: On Einstein Riemannian $g$-natural metrics on unit tangent sphere bundles. Ann. Global Anal. Geom. 38 (2010), 11-20. DOI 10.1007/s10455-010-9197-1 | MR 2657839 | Zbl 1206.53049
[8] M. T. K. Abbassi, M. Sarih: Killing vector fields on tangent bundles with Cheeger-Gromoll metric. Tsukuba J. Math. 27 (2003), 295-306. DOI 10.21099/tkbjm/1496164650 | MR 2025729 | Zbl 1060.53019
[9] M. T. K. Abbassi, M. Sarih: On natural metrics on tangent bundles of Riemannian manifolds. Arch. Math., Brno 41 (2005), 71-92. MR 2142144 | Zbl 1114.53015
[10] M. T. K. Abbassi, M. Sarih: On some hereditary properties of Riemannian $g$-natural metrics on tangent bundles of Riemannian manifolds. Differ. Geom. Appl. 22 (2005), 19-47. DOI 10.1016/j.difgeo.2004.07.003 | MR 2106375 | Zbl 1068.53016
[11] M. Benyounes, E. Loubeau, L. Todjihounde: Harmonic maps and Kaluza-Klein metrics on spheres. Rocky Mt. J. Math. 42 (2012), 791-821. DOI 10.1216/RMJ-2012-42-3-791 | MR 2966473 | Zbl 1257.58008
[12] G. Calvaruso, V. Martín-Molina: Paracontact metric structures on the unit tangent sphere bundle. Ann. Mat. Pura Appl. (4) 194 (2015), 1359-1380. DOI 10.1007/s10231-014-0424-4 | MR 3383943 | Zbl 1328.53035
[13] G. Calvaruso, D. Perrone: Homogeneous and $H$-contact unit tangent sphere bundles. J. Aust. Math. Soc. 88 (2010), 323-337. DOI 10.1017/S1446788710000157 | MR 2661453 | Zbl 1195.53045
[14] G. Calvaruso, D. Perrone: Metrics of Kaluza-Klein type on the anti-de Sitter space $\mathbb{H}_1^3$. Math. Nachr. 287 (2014), 885-902. DOI 10.1002/mana.201200105 | MR 3219219 | Zbl 1304.53070
[15] S. Deshmukh: Geometry of conformal vector fields. Arab J. Math. Sci. 23 (2017), 44-73. DOI 10.1016/j.ajmsc.2016.09.003 | MR 3589499 | Zbl 1356.53043
[16] P. Dombrowski: On the geometry of the tangent bundle. J. Reine Angew. Math. 210 (1962), 73-88. DOI 10.1515/crll.1962.210.73 | MR 141050 | Zbl 0105.16002
[17] S. Ewert-Krzemieniewski: On Killing vector fields on a tangent bundle with $g$-natural metric. I. Note Mat. 34 (2014), 107-133. DOI 10.1285/i15900932v34n2p107 | MR 3315987 | Zbl 1319.53017
[18] A. Gezer, L. Bilen: On infinitesimal conformal transformations with respect to the Cheeger-Gromoll metric. An. Ştiinţ. Univ. "Ovidius" Constanţa, Ser. Mat. 20 (2012), 113-128. DOI 10.2478/v10309-012-0009-4 | MR 2928413 | Zbl 1274.53027
[19] G. S. Hall: Symmetries and Curvature Structure in General Relativity. World Scientific Lecture Notes in Physics 46, World Scientific, River Edge (2004). DOI 10.1142/1729 | MR 2109072 | Zbl 1054.83001
[20] S. Hedayatian, B. Bidabad: Conformal vector fields on tangent bundle of a Riemannian manifold. Iran. J. Sci. Technol, Trans. A, Sci. 29 (2005), 531-539. MR 2239754 | Zbl 1106.53012
[21] I. Kolář, P. W. Michor, J. Slovák: Natural Operations in Differential Geometry. Springer, Berlin (1993). DOI 10.1007/978-3-662-02950-3 | MR 1202431 | Zbl 0782.53013
[22] T. Konno: Killing vector fields on tangent sphere bundles. Kodai Math. J. 21 (1998), 61-72. DOI 10.2996/kmj/1138043835 | MR 1625061 | Zbl 0907.53033
[23] T. Konno: Infinitesimal isometries on tangent sphere bundles over three-dimensional manifolds. Hiroshima Math. J. 41 (2011), 343-366. DOI 10.32917/hmj/1323700039 | MR 2895285 | Zbl 1235.53051
[24] O. Kowalski, M. Sekizawa: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles - a classification. Bull. Tokyo Gakugei Univ., Sect. IV, Ser. Math. Nat. Sci. 40 (1988), 1-29. MR 0974641 | Zbl 0656.53021
[25] E. Peyghan, A. Heydari: Conformal vector fields on tangent bundle of a Riemannian manifold. J. Math. Anal. Appl. 347 (2008), 136-142. DOI 10.1016/j.jmaa.2008.05.092 | MR 2433831 | Zbl 1145.53028
[26] S. Sasaki: On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J., II. Ser. 10 (1958), 338-354. DOI 10.2748/tmj/1178244668 | MR 0112152 | Zbl 0086.15003
[27] S. Sasaki: On the differential geometry of tangent bundles of Riemannian manifolds. II. Tohoku Math. J., II. Ser. 14 (1962), 146-155. DOI 10.2748/tmj/1178244169 | MR 0145456 | Zbl 0109.40505
[28] F. M. Şimşir, C. Tezer: Conformal vector fields with respect to the Sasaki metric tensor. J. Geom. 84 (2005), 133-151. DOI 10.1007/s00022-005-0033-x | MR 2215371 | Zbl 1093.53022
[29] S. Tanno: Killing vectors and geodesic flow vectors on tangent bundles. J. Reine Angew. Math. 282 (1976), 162-171. DOI 10.1515/crll.1976.282.162 | MR 0405285 | Zbl 0317.53042
[30] C. M. Wood: An existence theorem for harmonic sections. Manuscr. Math. 68 (1990), 69-75. DOI 10.1007/BF02568751 | MR 1057077 | Zbl 0713.58010
[31] K. Yano, S. Kobayashi: Prolongations of tensor fields and connections to tangent bundles. I: General theory. J. Math. Soc. Japan 18 (1966), 194-210. DOI 10.2969/jmsj/01820194 | MR 0193596 | Zbl 0141.38702
[32] K. Yano, S. Kobayashi: Prolongations of tensor fields and connections to tangent bundles. II: Infinitesimal automorphisms. J. Math. Soc. Japan 18 (1966), 236-246. DOI 10.2969/jmsj/01830236 | MR 0200857 | Zbl 0147.21501
Affiliations: Mohamed Tahar Kadaoui Abbassi (corresponding author), Noura Amri, Laboratory of Mathematical Sciences and Applications, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdallah, B.P. 1796, Fès-Atlas, Fez, Morocco, e-mail: mtk_abbassi@Yahoo.fr, amri.noura1992@gmail.com
