[en] 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 symplectic 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. (author)