Some calculations on Kaluza-Klein metric with respect to lifts in tangent bundle
Abstract
In the present paper, a Riemannian metric on the tangent bundle, which is another generalization of Cheeger-Gromoll metric and Sasaki metric, is considered. This metric is known as Kaluza-Klein metric in literature which is completely determined by its action on vector fields of type X^{H} and Y^{V}. We obtain the covarient and Lie derivatives applied to the Kaluza-Klein metric with respect to the horizontal and vertical lifts of vector fields, respectively on tangent bundle TM.
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References
M. T. K. Abbassi and M. Sarih, On some hereditary properties of Riemannian g− natural metrics on tangent bundles of Riemannian manifolds. Differential Geom. Appl. 22(1). (2005), 19-47. https://doi.org/10.1016/j.difgeo.2004.07.003
M. T. K. Abbassi and M. Sarih, Killing vector fields on tangent bundles with Cheeger-Gromoll metrics. Tsukuba J. Math. 27, (2003), 295-306. https://doi.org/10.21099/tkbjm/1496164650
M. Anastasiei, Locally conformal Kaehler structures on tangent bundle of a space form. Libertas Math. 19, (1999), 71-76.
M. Benyounes, E. Loubeau and L. K. Todjihounde, Harmonic maps and Kaluza-Klein metrics on spheres. Rocky Mount. J. of Math. 42(3), (2012), 791-821. https://doi.org/10.1216/RMJ-2012-42-3-791
J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature. Ann. of Math. 96, (1972), 413-443. https://doi.org/10.2307/1970819
A. Gezer and C. Karaman, Golden-Hessian structures. Proc. Nat. Acad. Sci. India Sect. A. 86(1), (2016), 41-46. https://doi.org/10.1007/s40010-015-0226-0
A. Gezer and M. Altunbas, Some notes concerning Riemannian metrics of Cheeger-Gromoll type. Jour. Math Anal. App. 396, (2012), 119-132. https://doi.org/10.1016/j.jmaa.2012.06.011
S. Gudmundsson and E. Kappos, On the Geometry of the Tangent Bundles. Expo. Math. 20, (2002), 1-41. https://doi.org/10.1016/S0723-0869(02)80027-5
Z. H. Hou and L. Sun, Geometry of Tangent Bundle with Cheeger-Gromoll type metric. Jour. Math. Anal. App. 402, (2013), 493-504. https://doi.org/10.1016/j.jmaa.2013.01.043
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry-Volume I,John Wiley & Sons, Inc, New York, 1963.
M. I. Munteanu, Some aspects on the geometry of the tangent bundles and tangent sphere bundles of Riemannian manifold. Mediterr. J. Math. 5(1), (2008), 43-59. https://doi.org/10.1007/s00009-008-0135-4
Z. Olszak, On almost complex structures with Norden metrics on tangent bundles. Period. Math. Hungar. 51(2), (2005), 59-74. https://doi.org/10.1007/s10998-005-0030-8
A. A. Salimov, Tensor Operators and Their applications, Nova Science Publ, New York, 2013.
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10, (1958), 338-358. https://doi.org/10.2748/tmj/1178244668
M. Sekizawa, Curvatures of tangent bundles with Cheeger-Gromoll metric. Tokyo J. of Math. 14(2), (1991), 407-417. https://doi.org/10.3836/tjm/1270130381
A. A. Salimov, A. Gezer and M. Iscan, On para-K¨ahler-Norden structures on the tangent bundles. Ann. Polon. Math. 103(3), (2012), 247-261. https://doi.org/10.4064/ap103-3-3
K. Yano and S. Ishihara, Tangent and cotangent bundles. Marcel Dekker. Inc. 1973.
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