As far as I understand it there are essentially two ways in which you can study quantum mechanics on a manifold with some curvature. Classically speaking these two ways lead to the same physics but in a quantum mechanical approach they are distinct.
The first approach is to think of a particle moving "freely" through three-dimensional space, but subject to external forces that confine the particle to some submanifold. The particle lives, in some sense, in a confining potential which defines the manifold. The phase space of the particle is, from the start, the usual phase space associated with the three-dimensional space. However, the external potential limits the particle to some subspace of this phase space.
The second approach is to work with generalized coordinates, as is done in Lagrangian mechanics. The coordinates of the particle are then a parametrization of the submanifold. What's important here is that there is no reference to the coordinates of the three-dimensional space. An example is the pendulum, which can be described solely in terms of the angle the pendulum makes with the z-axis.
Classically there is no distinction between the two approaches. This no longer holds when you move to quantum mechanics. If you follow the first approach, using some confining potential to keep the particle on the manifold, you will deal with the uncertainty principle that prohibits the exact localization of the particle onto the manifold. Because of this principle the particle will never be fully screened from the larger dimensional space. You can still systematically set up the quantization procedure, though. The advantage of this approach is that quantization works in the usual way (you work with cartesian coordinates, after all). The resolution is to essentially split up the wavefunction and the Schroedinger equation (S.E.) in contributions due to the confining potential and a sort-of effective S.E. for the remaining part of the wavefunction. The effective S.E. then contains two effective potentials due to the Mean curvature and Gauss curvature of the corresponding manifold.
This is a very important feature: a cylinder, for instance, has no Gauss curvature, only a mean curvature. In the second approach you will find that there is no distinction between two cylinders with different mean curvatures, because in this approach only the Gauss curvature pops up. Take for instance a particle living on a 1D line. 795
You only require one coordinate to describe this line, so for the the second approach all systems are equivalent. But in the first approach you have to specify in what way the line is embedded in the higher dimensional space, and how the particle is confined to the lower-dimensional space.
The second approach might feel more natural, if you think like a mathematician. In this approach you require a way to quantize generalized coordinates -- which is a lot more subtle than ordinary quantization. The problem that plagues this approach is the so-called ordering problem. Essentially you want to replace the momentum label by a derivation operator $p \rightarrow -i\hbar\nabla$. Furthermore, there's also the choice of parametrization of the manifold, which should ofcourse have no effect on the underlying physics (similar to general relativity). The ordering problem states that you do not know a priori which way the classical (commuting) variables have to be ordered before you replace them by their quantum mechanical (non-commuting) counterparts. What's even worse, because of the curvature of the space the derivative operator also contains some ambiguity. There is an ambiguity in the choice of your momentum operator and your Hamiltonian (and any other functions). Many quantum mechnical Hamiltonians have the same classical limit, and the equivalence principle (i.e. linking quantum mechanics to classical physics) does not dictate which is best. For instance, the kinetic operator $\nabla^2$ can be defined using the canonical Laplacian or the Laplace-Beltrami operator. Still, there is some work out there which motivates a generalized equivalence principle (see e.g. <PERSON>) and results in a consistent quantization procedure.
Both approaches have interesting feature, but the first one is actually a bit more physical. The reason is that in condensed matter you deal with confining potentials due to some ionic lattice. Take for instance graphene, which is a two-dimensional surface. As it turns out, this surface is not completely flat but will always form some ripples. These deformations of the surface can be interpreted as if the electrons (or Dirac fermions, if you want to use the effective theory) live on a curved manifold embedded in a three dimensional surface. This leads to hilarious applications, such as the existence of wormholes in Graphene. But in the end the curvature has a very physical manifestation in the electronic properties of the system. 976