CURVATURE PROPERTIES OF RIEMANNIAN METRICS OF THE FORM Sgf +H g ON THE TANGENT BUNDLE OVER A RIEMANNIAN MANIFOLD (M,g)

Abstract

In this paper, we define a special new family of metrics which rescale the horizontal part by a nonzero differentiable function on the tangent bundle over a Riemannian manifold. We investigate curvature properties of the Levi-Civita connection and another metric connection of the new Riemannian metric.

Statistical structures and Killing vector fields on tangent bundles with respect to two different metrics

“…However, some other metrics can be defined on the tangent bundle which are not subclasses of this g−natural metric. As first example, in [9], Gezer and Ozkan defined a metric G f 1 = c g + v (f g), where c g is the complete lift of the metric and v (f g) is the vertical lift of f g and f is a smooth function on M. As second example, in [8], Gezer et al introduced a metric G f 2 = s g f + h g, where s g f is a metric which is obtained by rescaling the Sasaki metric with a smooth function f on M and h g is the horizontal lift of g. These lifts will be explained later and we will deal with these two metrics in this paper.…”

On the geometry of tangent bundle of a hypersurface in Rn+1

“…This problem in the sasaki metric has led researchers to search for other metrics on the tangent bundle for example Cheeger-Gromoll metric, Oproiu metric (cf. [1,3,[7][8][9][10][11]). Since the projection π : T M → M is a Riemannian submersion, all metrics defined on T M are appropriate with this smooth projection.…”