I do not understand well the question. Are you asking whether the issue of (first) quantization on manifolds has been investigated? If this is the question the answer is positive. Several researchers focused on that problem. The point is that the standard procedure based on Stone-von Neumann theorem generally does not hold for manifolds different form ${\mathbb R}^n$. The algebra of elementary observables (that generated by position and momentum in $\mathbb R^n$ is very difficult to define). Naive approaches face the technical problem of symmetric operators which should represent observables but are not (essentially) self adjoint. The situation becomes simpler when the space is homogeneous, i.e. when there is a (at least topological) group acting transitively on the space. The action can be defined either in terms of isometries, conformal transformations or diffeomorphisms. In this case there are procedures, in particular due to Isham and Landsman and collaborators (Letters in Mathemattical Physics 20:11-18, 1990, Nuclear Physics B365 (1991) 121-160) leading to a *-algebra of essentially self adjoint operators defining observables in a suitable Hilbert space associated with the manifold. These procedures include (generalizations of) anyons theory and Aharonov Bohm quantization as well as the standard quantization procedure in $\mathbb R^n$.