Some notes concerning Riemannian metrics of Cheeger Gromoll type

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