  The effects of surface curvature on the motion of electrons in a mesoscopic
two-dimensional ring on a cone in the presence of external magnetic fields are
examined. The approach follows the thin-layer quantization procedure, which
gives rise to a geometry induced potential. Due to the annular geometric shape
of the sample, only the mean curvature has relevant effects to the model.
Nevertheless, the most significant contribution of the mean curvature occurs in
the state $m=0$, which tends to decrease the energies when the magnetic field
is null. The effects of curvature are also manifested in the cyclotron
frequency as well as in the effective angular momentum through the $\alpha$
parameter, which can be controlled in such a way that the magnitude of these
effects becomes explicit. This is verified in the energies and wave functions
of the system. A decrease in the number of occupied states in the Fermi energy
is observed. As a consequence, there is an alteration in the radial range of
the conducting region of the sample. This fact is confirmed by studying the
variations in the radii of the states.
