  By considering a cigar-shaped trapping potential elongated in a proper
curvilinear coordinate, we discover a new form of wave localization which
arises from the interplay of geometry and topological protection. The potential
is modulated in its shape such that local curvature introduces a trapping
potential. The curvature varies along the trap curvilinear axis encodes a
topological Harper modulation. The varying geometry maps our system in a
one-dimensional Andre-Aubry-Harper grating. We show that a mobility edge exists
and topologically protected states arises. These states are extremely robust
with respect to disorder in shape of the string. The results may be relevant
for localization phenomena in Bose-Einstein condensates, optical fibers and
waveguides, and new laser devices, but also for fundamental studies on string
theory. Taking into account that the one-dimensional modulation mimic the
existence of a additional dimensions, our system is the first example of
physically realizable five-dimensional string.
