The geometry of generalized Cheeger-Gromoll metrics on the total space of transitive Euclidean Lie algebroids

Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family h_p,q of natural metrics on the tangent bundle depending on two parameters with p ∈ R and q ≥ 0. This family possesses interesting geometric properties. If p = q = 0 we recover the Sasaki metric and when p = q = 1 we recover the classical Cheeger-Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with a Euclidean product on its total space. In this paper, we show that the analogous of natural metrics