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). (C) 2012 Elsevier Inc. All rights reserved.