  Quantum particles confined to surfaces in higher dimensional spaces are acted
upon by forces that exist only as a result of the surface geometry and the
quantum mechanical nature of the system. The dynamics are particularly rich
when confinement is implemented by forces that act normal to the surface. We
review this confining potential formalism applied to the confinement of a
particle to an arbitrary manifold embedded in a higher dimensional Euclidean
space. We devote special attention to the geometrically induced gauge potential
that appears in the effective Hamiltonian for motion on the surface. We
emphasize that the gauge potential is only present when the space of states
describing the degrees of freedom normal to the surface is degenerate. We also
distinguish between the effects of the intrinsic and extrinsic geometry on the
effective Hamiltonian and provide simple expressions for the induced scalar
potential. We discuss examples including the case of a 3-dimensional manifold
embedded in a 5-dimensional Euclidean space.
