  The geometric effects of two-dimensional curved systems have been an
interesting topic for a long time. A M\"{o}bius surface is specifically
considered. For a relativistic particle confined to the nontrivial surface, we
give the effective Dirac equation in the thin-layer quantization formalism, and
we find a geometric gauge potential that results from the rotation
transformation of the local frame moving on M\"obius strip, and an effective
mass that is from the rescaling transformation. Intriguingly, the geometric
gauge potential can play a role of monopole magnetic field for the particles
with spin, and which can produce quantum spin Hall effects. As potential
applications, effective monopole magnetic fields and spin Hall phenomena can be
generated and manipulated by designing the geometries and topologies of
two-dimensional nanodevices.

Quantum Physics;Mesoscale and Nanoscale Physics