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

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 $(M_n,g)$.

Metallic Riemannian Structures on the Tangent Bundles of Riemannian Manifolds with $g-$Natural Metrics

“…Let (M, g) be a Riemannian manifold and T M be its tangent bundle. In [4], Abbassi and Sarih defined g−natural metrics on T M as metrics which arise from g through first order natural operators defined between the natural bundle of Riemannian metrics on M and the natural bundle of (0, 2)−tensor fields on T M. Some well-known examples of g−natural metrics are the Sasaki metric ( [8], [20]), Sasaki type metrics ( [9]), the Cheeger-Gromoll metric ( [19], [21]), Cheeger-Gromoll type metrics ( [7], [10]) and the Kaluza-Klein metric ( [6]). Abbassi et al have been studied geometric properties of tangent bundles with respect to g−natural metrics (see [1], [2], [3], for instance).…”