g-natural metrics: new horizons in the geometry of tangent bundles of Riemannian manifolds

Traditionally, the Riemannian geometry of tangent and unit tangent bundles was related to the Sasaki metric. The study of the relationship between the geometry of a manifold (M,g) and that of its tangent bundle TM equipped with the Sasaki metric gs had shown some kinds of rigidity. The concept of naturality allowed O.Kowalski and M.Sekizawa to introduce a wide class of metrics on TM naturally constructed from some classical and non-classical lifts of g. This class contains the Sasaki metric as well as the well known Cheeger-Gromoll metric and the metrics of Oproiu-type. We review some of the most interesting results, obtained recently, concerning the geometry of the tangent and the unit tangent bundles equipped with an arbitrary Riemannian g-natural metric