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

Edge Majoranas on locally flat surfaces - the cone and the M\"obius band