  We define a class of metrics that extend the Sasaki metric of a tangent
manifold of a Riemannian manifold. The new metrics are obtained by the transfer
of the generalized (pseudo-)Riemannian metrics of the pullback of the big
tangent bundle of a manifold to the tangent manifold. We obtain the expression
of the transferred metric and we define a canonical, metric connection with
torsion. We calculate the torsion, curvature and Ricci curvature of this
connection and give a few applications of the results. We also discuss the
transfer of generalized complex and generalized Kaehler structures.
