The formulation of Geometric Quantization contains several ax- ioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introduc- ing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liou- ville symplectic manifolds. Our main application of this modified geo- metric 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.
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