  The formulation of Geometric Quantization contains several axioms and
assumptions. We show that for real polarizations we can generalize the standard
geometric quantization procedure by introducing an arbitrary connection on the
polarization bundle. The existence of reducible quantum structures leads to
considering the class of Liouville symplectic manifolds. Our main application
of this modified geometric quantization scheme is to Quantum Mechanics on
Riemannian manifolds. With this method we obtain an energy operator without the
scalar curvature term that appears in the standard formulation, thus agreeing
with the usual expression found in the Physics literature.
