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.