  Studies of the formation of Landau levels based on the Schr\"odinger equation
for electrons constrained to curved surfaces have a long history. These include
as prime examples surfaces with constant positive and negative curvature, the
sphere [Phys. Rev. Lett. 51, 605 (1983)] and the pseudosphere [Annals of
Physics 173, 185 (1987)]. Now, topological insulators, hosting Dirac-type
surface states, provide a unique platform to experimentally examine such
quantum Hall physics in curved space. Hence, extending previous work we
consider solutions of the Dirac equation for the pseudosphere for both, the
case of an overall perpendicular magnetic field and a homogeneous coaxial,
thereby locally varying, magnetic field. For both magnetic-field
configurations, we provide analytical solutions for spectra and eigenstates.
For the experimentally relevant case of a coaxial magnetic field we find that
the Landau levels split and show a peculiar scaling $\propto B^{1/4}$, thereby
characteristically differing from the usual linear $B$ and $B^{1/2}$ dependence
of the planar Schr\"odinger and Dirac case, respectively. We compare our
analytical findings to numerical results that we also extend to the case of the
Minding surface.
