ON NATURAL METRICS ON TANGENT BUNDLES OF RIEMANNIAN MANIFOLDS

There is a class of metrics on the tangent bundle TM of a Riemannian manifold (M, g) (oriented, or non-oriented, respectively), which are 'naturally constructed' from the base metric g [15]. We call them "g-natural metrics" on TM. To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. [18]) of finding Riemannian metrics on TM from some quadratic forms on OM x R-m to find metrics (not necessary Riemannian) on TM, we prove that all g-natural metrics on TM can be obtained by Musso-Tricerri's generalized scheme. We calculate also the Levi-Civita connection of Riemannian g-natural metrics on TM. As application, we sort out all Riemannian g-natural metrics with the following properties, respectively: 1) The fibers of TM are totally geodesic. 2) The geodesic flow on TM is incompressible. We shall limit ourselves to the non-oriented situation.