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

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.