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

TL;DR

This paper studies the curvature and geometric structures of the cotangent bundle of a Riemannian manifold equipped with a Cheeger Gromoll type metric, focusing on conditions for special structures like para-K"ahler and quasi-K"ahler.

Contribution

It introduces and analyzes a rescaled Cheeger Gromoll type metric on the cotangent bundle, exploring its curvature and almost paracomplex Norden structures.

Findings

Conditions for para-K"ahler and quasi-K"ahler structures are established.

Curvature properties of the cotangent bundle with the rescaled metric are characterized.

Properties of almost paracomplex Norden structures are related to almost product Riemannian manifolds.

Abstract

Let $(M,g)$ be an n-dimensional Riemannian manifold and $T^{*}M$ 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 $T^{*}M$ and construct almost paracomplex Norden structures on $T^{*}M$. 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.