On the Geometry of the Tangent Bundle With Vertical Rescaled Metric

Let (M,g) be a n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by Gᶠ and called the vertical rescaled metric on the tangent bundle TM. We calculate its Levi-Civita connection and Riemannian curvature tensor. We study the geometry of (TM,Gᶠ) and several important results are obtained on curvature, Einstein structure, scalar and sectional curvatures