| Title:
|
On $g$-natural conformal vector fields on unit tangent bundles (English) |
| Author:
|
Abbassi, Mohamed Tahar Kadaoui |
| Author:
|
Amri, Noura |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
75-109 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
conformal vector field |
| Keyword:
|
unit tangent bundle |
| Keyword:
|
$g$-natural metric |
| MSC:
|
53C07 |
| MSC:
|
53C24 |
| MSC:
|
53C25 |
| idZBL:
|
07332707 |
| idMR:
|
MR4226472 |
| DOI:
|
10.21136/CMJ.2020.0193-19 |
| . |
| Date available:
|
2021-03-12T16:10:28Z |
| Last updated:
|
2023-04-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148730 |
| . |
| Reference:
|
[1] Abbassi, M. T. K.: Métriques naturelles sur le fibré tangent à une variété Riemannienne.Editions universitaires europeennes, Paris (2012), French. |
| Reference:
|
[2] Abbassi, M. T. K., Amri, N., Calvaruso, G.: Kaluza-Klein type Ricci solitons on unit tangent sphere bundles.Differ. Geom. Appl. 59 (2018), 184-203. Zbl 1391.53053, MR 3804828, 10.1016/j.difgeo.2018.04.006 |
| Reference:
|
[3] Abbassi, M. T. K., Calvaruso, G.: $g$-natural contact metrics on unit tangent sphere bundles.Monaths. Math. 151 (2007), 89-109. Zbl 1128.53049, MR 2322938, 10.1007/s00605-006-0421-9 |
| Reference:
|
[4] Abbassi, M. T. K., Calvaruso, G.: $g$-natural metrics of constant curvature on unit tangent sphere bundles.Arch. Math., Brno 48 (2012), 81-95. Zbl 1274.53112, MR 2946208, 10.5817/AM2012-2-81 |
| Reference:
|
[5] Abbassi, M. T. K., Kowalski, O.: 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. Zbl 1107.53023, MR 2181488, 10.1142/9789812701701_0001 |
| Reference:
|
[6] Abbassi, M. T. K., Kowalski, O.: Naturality of homogeneous metrics on Stiefel manifolds $SO(m+1)/SO(m-1)$.Differ. Geom. Appl. 28 (2010), 131-139. Zbl 1190.53020, MR 2594457, 10.1016/j.difgeo.2009.05.007 |
| Reference:
|
[7] Abbassi, M. T. K., Kowalski, O.: On Einstein Riemannian $g$-natural metrics on unit tangent sphere bundles.Ann. Global Anal. Geom. 38 (2010), 11-20. Zbl 1206.53049, MR 2657839, 10.1007/s10455-010-9197-1 |
| Reference:
|
[8] Abbassi, M. T. K., Sarih, M.: Killing vector fields on tangent bundles with Cheeger-Gromoll metric.Tsukuba J. Math. 27 (2003), 295-306. Zbl 1060.53019, MR 2025729, 10.21099/tkbjm/1496164650 |
| Reference:
|
[9] Abbassi, M. T. K., Sarih, M.: On natural metrics on tangent bundles of Riemannian manifolds.Arch. Math., Brno 41 (2005), 71-92. Zbl 1114.53015, MR 2142144 |
| Reference:
|
[10] Abbassi, M. T. K., Sarih, M.: On some hereditary properties of Riemannian $g$-natural metrics on tangent bundles of Riemannian manifolds.Differ. Geom. Appl. 22 (2005), 19-47. Zbl 1068.53016, MR 2106375, 10.1016/j.difgeo.2004.07.003 |
| Reference:
|
[11] Benyounes, M., Loubeau, E., Todjihounde, L.: Harmonic maps and Kaluza-Klein metrics on spheres.Rocky Mt. J. Math. 42 (2012), 791-821. Zbl 1257.58008, MR 2966473, 10.1216/RMJ-2012-42-3-791 |
| Reference:
|
[12] Calvaruso, G., Martín-Molina, V.: Paracontact metric structures on the unit tangent sphere bundle.Ann. Mat. Pura Appl. (4) 194 (2015), 1359-1380. Zbl 1328.53035, MR 3383943, 10.1007/s10231-014-0424-4 |
| Reference:
|
[13] Calvaruso, G., Perrone, D.: Homogeneous and $H$-contact unit tangent sphere bundles.J. Aust. Math. Soc. 88 (2010), 323-337. Zbl 1195.53045, MR 2661453, 10.1017/S1446788710000157 |
| Reference:
|
[14] Calvaruso, G., Perrone, D.: Metrics of Kaluza-Klein type on the anti-de Sitter space $\mathbb{H}_1^3$.Math. Nachr. 287 (2014), 885-902. Zbl 1304.53070, MR 3219219, 10.1002/mana.201200105 |
| Reference:
|
[15] Deshmukh, S.: Geometry of conformal vector fields.Arab J. Math. Sci. 23 (2017), 44-73. Zbl 1356.53043, MR 3589499, 10.1016/j.ajmsc.2016.09.003 |
| Reference:
|
[16] Dombrowski, P.: On the geometry of the tangent bundle.J. Reine Angew. Math. 210 (1962), 73-88. Zbl 0105.16002, MR 141050, 10.1515/crll.1962.210.73 |
| Reference:
|
[17] Ewert-Krzemieniewski, S.: On Killing vector fields on a tangent bundle with $g$-natural metric. I.Note Mat. 34 (2014), 107-133. Zbl 1319.53017, MR 3315987, 10.1285/i15900932v34n2p107 |
| Reference:
|
[18] Gezer, A., Bilen, L.: On infinitesimal conformal transformations with respect to the Cheeger-Gromoll metric.An. Ştiinţ. Univ. ``Ovidius'' Constanţa, Ser. Mat. 20 (2012), 113-128. Zbl 1274.53027, MR 2928413, 10.2478/v10309-012-0009-4 |
| Reference:
|
[19] Hall, G. S.: Symmetries and Curvature Structure in General Relativity.World Scientific Lecture Notes in Physics 46, World Scientific, River Edge (2004). Zbl 1054.83001, MR 2109072, 10.1142/1729 |
| Reference:
|
[20] Hedayatian, S., Bidabad, B.: Conformal vector fields on tangent bundle of a Riemannian manifold.Iran. J. Sci. Technol, Trans. A, Sci. 29 (2005), 531-539. Zbl 1106.53012, MR 2239754 |
| Reference:
|
[21] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry.Springer, Berlin (1993). Zbl 0782.53013, MR 1202431, 10.1007/978-3-662-02950-3 |
| Reference:
|
[22] Konno, T.: Killing vector fields on tangent sphere bundles.Kodai Math. J. 21 (1998), 61-72. Zbl 0907.53033, MR 1625061, 10.2996/kmj/1138043835 |
| Reference:
|
[23] Konno, T.: Infinitesimal isometries on tangent sphere bundles over three-dimensional manifolds.Hiroshima Math. J. 41 (2011), 343-366. Zbl 1235.53051, MR 2895285, 10.32917/hmj/1323700039 |
| Reference:
|
[24] Kowalski, O., Sekizawa, M.: 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. Zbl 0656.53021, MR 0974641 |
| Reference:
|
[25] Peyghan, E., Heydari, A.: Conformal vector fields on tangent bundle of a Riemannian manifold.J. Math. Anal. Appl. 347 (2008), 136-142. Zbl 1145.53028, MR 2433831, 10.1016/j.jmaa.2008.05.092 |
| Reference:
|
[26] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds.Tohoku Math. J., II. Ser. 10 (1958), 338-354. Zbl 0086.15003, MR 0112152, 10.2748/tmj/1178244668 |
| Reference:
|
[27] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds. II.Tohoku Math. J., II. Ser. 14 (1962), 146-155. Zbl 0109.40505, MR 0145456, 10.2748/tmj/1178244169 |
| Reference:
|
[28] Şimşir, F. M., Tezer, C.: Conformal vector fields with respect to the Sasaki metric tensor.J. Geom. 84 (2005), 133-151. Zbl 1093.53022, MR 2215371, 10.1007/s00022-005-0033-x |
| Reference:
|
[29] Tanno, S.: Killing vectors and geodesic flow vectors on tangent bundles.J. Reine Angew. Math. 282 (1976), 162-171. Zbl 0317.53042, MR 0405285, 10.1515/crll.1976.282.162 |
| Reference:
|
[30] Wood, C. M.: An existence theorem for harmonic sections.Manuscr. Math. 68 (1990), 69-75. Zbl 0713.58010, MR 1057077, 10.1007/BF02568751 |
| Reference:
|
[31] Yano, K., Kobayashi, S.: Prolongations of tensor fields and connections to tangent bundles. I: General theory.J. Math. Soc. Japan 18 (1966), 194-210. Zbl 0141.38702, MR 0193596, 10.2969/jmsj/01820194 |
| Reference:
|
[32] Yano, K., Kobayashi, S.: Prolongations of tensor fields and connections to tangent bundles. II: Infinitesimal automorphisms.J. Math. Soc. Japan 18 (1966), 236-246. Zbl 0147.21501, MR 0200857, 10.2969/jmsj/01830236 |
| . |
