  We investigate the effects of the curved geometry on a massless relativistic
electron constrained to a graphene strip with a Moebius strip shape. The
anisotropic and parity-violating geometry of the Moebius band produces a
geometric potential that inherits these features. By considering wires along
the strip width and the strip length, we find exact solutions for the Dirac
equation and the effects of the geometric potential on the electron were
explored. In both cases, the geometric potential yields to a geometric phase on
the wave function. Along the strip width, the density of states depends on the
direction chosen for the wire, a consequence of the lack of axial symmetry.
Moreover, the breaking of the parity symmetry enables the electronic states to
be concentrated on the inner or on the outer portion of the strip. For wires
along the strip length, the nontrivial topology influences the eigenfunctions
by modifying their periodicity. It turns out that the ground state has a period
of $4\pi$ whereas the first excited state is a $2\pi$ periodic function.
Moreover, we found that the energy levels are half-integer multiples of the
energy of the ground state.
