Notes on the tangent bundle with deformed complete lift metric

Abstract

In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric.

Derivatives with respect to horizontal and vertical lifts of the deformed complete lift metric G_{f} on tangent bundle.

“…In this section, we give the Levi-Civita connection ∇ of the tangent bundle T M with the deformed complete lift metric Gf and study fiber-preserving Killing vector fields on T M . The deformed complete lift metric Gf is defined by [6] G…”

“…for all vector fields X and Y on M , and f > 0, f ∈ C ∞ (M ) [6]. For f = 1, it follows that g = S g, i.e.…”

“…Let X = X i ∂ ∂x i be the local expressions in U of a vector field X on M . The vertical lift X V , the horizontal lift X H , and the complete lift X C of X are then given respectively by [6]…”

On the geometry of lift metrics and lift connections on the tangent bundle

“…The geometry of the tangent bundle with Riemannian lift metrics has been extensively studied in recent years (see [3], [6], [7], [9], [10], [20]).…”

Geometry of Mus-Sasaki metric

“…The geometry of the tangent bundle T M equiped with Sasaki metric has been studied by many authors Sasaki [16], K. Yano and S. Ishihara [18], P. Dombrowski [7], A. Salimov, A. Gezer and N. Cengiz (see [2], [8], [12], [13], [14]) etc. The rigidity of Sasaki metric has incited some geometers to construct and study other metrics on T M .…”