  Motivated by the transmission of topological surface states through atomic
scale steps, we study the transport of gapless Dirac fermions on hyperbolic
surfaces. We confirm that, independent of the curvature of the hyperbolae and
the sharpness of the corners, no backward scattering takes place and
transmission of the topological surface states is completely independent of the
geometrical shape (within the hyperbolic model) of the surface. The density of
states of the electrons, however, shows a dip at concave step edges which can
be measured by an STM tip. We also show that the tunneling conductance measured
by a polarized scanning tunneling probe exhibits an unconventional dependence
on the polar and azimuthal angles of the magnetization of the tip as a function
of the curvature of the surface and the sharpness of the edge.
