  In this paper, we investigate the edge Majorana modes in the simplest
possible $p{}_{x}+ip_{y}$ superconductor defined on surfaces with different
geometry - the annulus, the cylinder, the M\"obius band and a cone (by cone we
mean a cone with the tip cut away so it is topologically equivalent to the
annulus and cylinder)- and with different configuration of magnetic fluxes
threading holes in these surfaces. In particular, we shall address two
questions: Given that, in the absence of any flux, the ground state on the
annulus does not support Majorana modes, while the one on the cylinder does,
how is it possible that the conical geometry can interpolate smoothly between
the two? Given that in finite geometries edge Majorana modes have to come in
pairs, how can a $p{}_{x}+ip_{y}$ state be defined on a M\"obius band, which
has only one edge? We show that the key to answering these questions is that
the ground state depends on the geometry, even though all the surfaces are
locally flat. In the case of the truncated cone, there is a non-trivial
holonomy, while the non-orientable M\"obius band must necessarily support a
domain wall.
