  In this work we obtain the Landau levels and the Hall conductivity at zero
temperature of a two-dimensional electron gas on a conical surface. We
investigate the integer quantum Hall effect considering two different
approaches. The first one is an extrinsic approach which employs an effective
scalar potential that contains both the Gaussian and the mean curvature of the
surface. The second one, an intrinsic approach where the Gaussian curvature is
the sole term in the scalar curvature potential. From a theoretical point of
view, the singular Gaussian curvature of the cone may affect the wave functions
and the respective Landau levels. Since this problem requests {\it self-adjoint
extensions}, we investigate how the conical tip could influence the integer
quantum Hall effect, comparing with the case were the coupling between the wave
functions and the conical tip is ignored. This last case corresponds to the
so-called {\it Friedrichs extension}. In all cases, the Hall conductivity is
enhanced by the conical geometry depending on the opening angle. There are a
considerable number of theoretical papers concerned with the self-adjoint
extensions on a cone and now we hope the work addressed here inspires
experimental investigation on these questions about quantum dynamics on a cone.
