Some notes concerning Riemannian metrics of Cheeger Gromoll type

Let ( M , g )  be an n -dimensional Riemannian manifold and T M  its tangent bundle. The purpose of the present paper is three-fold. Firstly, to study paraholomorphy property of two Riemannian metrics g a and g a , b  of Cheeger Gromoll type depending on one parameter and two parameters by using compatible paracomplex structures J a and J a , b on the tangent bundle T M . Secondly, to classify Killing vector fields on the tangent bundle T M equipped with the Riemannian metric g a , b . Finally, to give a detailed description of geodesics on the tangent bundle T M with respect to the Riemannian metric g a , b .