On the rescaled Riemannian metric of Cheeger Gromoll type on the cotangent bundle

Let $(M,g)$ be an n-dimensional Riemannian manifold and $T^{\ast }M$ be its cotangent bundle equipped with a Riemannian metric of Cheeger Gromoll type which rescale the horizontal part by a nonzero differentiable function. The main purpose of the present paper is to discuss curvature properties of $T^{\ast }M$ and construct almost paracomplex Norden structures on $T^{\ast }M$. We investigate conditions for these structures to be para-K"ahler (paraholomorphic) and quasi-K"ahler. Also, some properties of almost paracomplex Norden structures in context of almost product Riemannian manifolds are presented.