Induced symplectic connection on the phase space

It is shown that the general theory of lifting the tensor fields from a Rie mannian manifold M to its tangent bundle TM enables one to define in a natu ral mariner the unique sympletic connection on the phase space T*M which is induced by the Levi-Civita connection on M. This is exactly the symplectic connection given also by Bordemann, Neumaier and Waldmann Commun. Math. Ph ys. 198, 363 (1998); J. Geom. Phys. 29, 199 (1999). Relationship between th e symplectic and Riemannian geometries on T*M and M is considered.