It is shown that the general theory of lifting the tensor fields from a Riemannian manifold M to its tangent bundle TM enables one to define in a natural manner 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. Phys. 198, 363 (1998); J. Geom. Phys. 29, 199 (1999). Relationship between the symplectic and Riemannian geometries on T^* M and M is considered.

PACS numbers: 02.40.Ky