  We prove that curvature effects in low-dimensional nanomaterials can promote
the generation of topological states of matter by considering the paradigmatic
example of quantum wires with Rashba spin-orbit coupling, which are
periodically corrugated at the nanometer scale. The effect of the periodic
curvature generally results in the appearance of insulating phases with a
corresponding novel butterfly spectrum characterized by the formation of fine
measure complex regions of forbidden energies. When the Fermi energy lies in
the gaps, the system displays localized end states protected by topology. We
further show that for certain corrugation periods the system possesses
topologically non-trivial insulating phases at half-filling. Our results
suggest that the local curvature and the topology of the electronic states are
inextricably intertwined in geometrically deformed nanomaterials.
