  The nonrelativistic quantum dynamics of a spinless charged particle in the
presence of the Aharonov--Bohm potential in curved space is considered. We
chose the surface as being a cone defined by a line element in polar
coordinates. The geometry of this line element establishes that the motion of
the particle can occur on the surface of a cone or an anti--cone. As a
consequence of the nontrivial topology of the cone and also because of
two--dimensional confinement, the geometric potential should be taken into
account. At first, we establish the conditions for the particle describing a
circular path in such a context. Because of the presence of the geometric
potential, which contains a singular term, we use the self--adjoint extension
method in order to describe the dynamics in all space including the
singularity. Expressions are obtained for the bound state energies and wave
functions.
