The paper considers a class of g-natural metrics G on the tangent bundle T M of a Riemannian manifold (M, g). We prove that the flatness of g is necessary and sufficient for a metric G to be weakly symmetric (recurrent or pseudo-symmetric). Also, it is shown that the weak symmetry and recurrent or pseudo-symmetry properties of Sasakian lift metric, studied by Bejan ́ and Crasmareanu, and Binh and Tam assy, respectively, are special cases of our result.
