  Considering the class G of g-natural metrics on the tangent bundle of a
Riemannian manifold (M, g), it is shown that the flatnees for g is a necessary
and sufficient condition of weakly symmetry (recurrent or pseudo-symmetry) of
G. In particular, the cases of weakly symmetric Sasakian lift metric studied by
Bejan and Crasmareanu and recurrent or pseudo-symmetric Sasakian lift metric
studied by Binh and Tamassy are obtained.
