IFHP Transformations on the Tangent Bundle with the Deformed Complete Lift Metric

Year 2022, , 153 - 159, 30.04.2022

Mosayeb Zohrehvand

https://doi.org/10.36890/iejg.1037651

Abstract

Let $(M_n,g)$ be a Riemannian manifold and $TM_n$ the total space of its tangent bundle. In this paper, we determine the infinitesimal fiber-preserving holomorphically projective (IFHP) transformations on $TM_n$ with respect to the Levi-Civita connection of the deformed complete lift metric $\tilde{G}_f=g^C+(fg)^V$, where $f$ is a nonzero differentiable function on $M_n$ and $g^C$ and $g^V$ are the complete lift and the vertical lift of $g$ on $TM_n$, respectively. Morevore, we prove that every IFHP transformation on $(TM_n,\tilde{G}_f)$ is reduced to an affine and induces an infinitesimal affine transformation on $(M_n,g)$.

Keywords

Complete lift metric, Infinitesimal fiber-preserving transformation, Infinitesimal holomorphically projective transformations, adapted almost complex structure

References

• [1] Abbassi, M.T.K., Sarih, M.: On natural metrics on tangent bundles of Riemannian manifolds. Arch. Math. (Brno) Tomus. 41, 71-92 (2005).
• [2] Abbassi, M.T.K., Sarih, M.: On Riemannian g-natural metrics of the form ags + bgh + cgv on the tangent bundle of a Riemannian manifold (M; g). Mediterr. J. Math. 2, 19-43 (2005).
• [3] Bejan, C.L., Dru¸t˘a-Romaniuc, S.L.: H-projectively Euclidean Kähler tangent bundles of natural diagonal type. Publ. Math. Debrecen. 89 (4), 499-511 (2016).
• [4] Gezer, A.: On infinitesimal conformal transformations of the tangent bundles with the synectic lift of a Riemannian metric. Proc. Indian Acad. Sci. (Math. Sci.) 119 (3), 345-350 (2009).
• [5] Gezer, A.: On infinitesimal holomorphically projective transformations on the tangent bundles with respect to the Sasaki metric. Proceedings of the Estonian Academy of Sciences. 60 (3), 149-157 (2011).
• [6] Gezer, A., Özkan, M.: Notes on the tangent bundle with deformed complete lift metric. Turkish Journal of Mathematics. 38, 1038-1049 (2014).
• [7] Hasegawa, I., Yamauchi, K.: Infinitesimal holomorphically projective transformations on the tangent bundles with horizontal lift connection and adapted almost complex structure. J. Hokkaido Univ. Educ. 53, 1-8 (2003).
• [8] Hasegawa, I., Yamauchi, K.: Infinitesimal holomorphically projective transformations on the tangent bundles with complete lift connection. Differential Geometry-Dynamical Systems. 7, 42-48 (2005).
• [9] Hasegawa, I., Yamauchi, K.: On infinitesimal holomorphically projective transformations in compact Kaehlerian manifolds. Hokkaido Math. J. 8, 214-219 (1979).
• [10] Ishihara, S.: Holomorphically projective changes and their groups in an almost complex manifold. Tohoku Math. J. 9, 273-297 (1957).
• [11] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10, 338-358 (1958).
• [12] Tachibana, S., Ishihara, S.: On infinitesimal holomorphically projective transformations in Kahlerian manifolds. Tohoku Math. J. 10, 77-101 (1960).
• [13] Tarakci, O., Gezer, A., Salimov, A. A.: On solutions of IHPT equations on tangent bundle with the metric II+III. Math. Comput. Modelling. 50, 953-958 (2009).
• [14] Yamauchi, K.: On infinitesimal conformal transformations of the tangent bundles over Riemannian manifolds. Ann. Rep. Asahikawa. Med. Coll. 16, 1-6 (1995).
• [15] Yano, K.: The Theory of Lie Derivatives and Its Applications. Bibliotheca mathematica, North Holland Pub. Co. (1957).
• [16] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker, Inc. New York (1973).
• [17] Yano, K., Kobayashi, S.: Prolongation of tensor fields and connections to tangent bundles I, II, III. J. Math. Soc. Japan. 18, 194-210, 236–246 (1966), 19, 486-488 (1967).
• [18] Zohrehvand, M.: IFHP transformations on the tangent bundle of a Riemannian manifold with a class of pseudo-Riemannian metrics. C. R. Acad. Bulg. Sci. 73 (2), 170-178 (2020).
• [19] Zohrehvand, M.: Projective vector fields on the tangent bundle with the deformed complete lift metrics. Balkan J. Geom. Appl. 25 (2), 170-178 (2020).