Some notes concerning Riemannian metrics of Cheeger Gromoll type

Let (M,g) be an n-dimensional Riemannian manifold and TM 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 TM. Secondly, to classify Killing vector fields on the tangent bundle TM equipped with the Riemannian metric g a, b. Finally, to give a detailed description of geodesics on the tangent bundle TM with respect to the Riemannian metric g a, b. © 2012 Elsevier Ltd.