  According to Dirac, fundamental laws of Classical Mechanics should be
recovered by means of an "appropriate limit" of Quantum Mechanics. In the same
spirit it is reasonable to enquire about the fundamental geometric structures
of Classical Mechanics which will survive the appropriate limit of Quantum
Mechanics. This is the case for the symplectic structure. On the contrary, such
geometric structures as the metric tensor and the complex structure, which are
necessary for the formulation of the Quantum theory, may not survive the
Classical limit, being not relevant in the Classical theory. Here we discuss
the Classical limit of those geometric structures mainly in the Ehrenfest and
Heisenberg pictures, i.e. at the level of observables rather than at the level
of states. A brief discussion of the fate of the complex structure in the
Quantum-Classical transition in the Schroedinger picture is also mentioned.
