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 TP 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 TM And M is considered.
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