Vertical rescaled Cheeger-Gromoll metric and harmonicity on the cotangent bundle

In this paper, we introduce the harmonicity on the cotangent bundle equipped with vertical rescaled Cheeger-Gromoll metric which rescales the vertical part by a non-zero differentiable function f. We establish a necessary and sufficient condition under which a covector field is harmonic with respect to this metric. We also construct some examples of harmonic vector fields. Finally, we study the harmonicity of a map between a Riemannian manifold and a cotangent bundle of another Riemannian manifold and vice versa.