  We investigate the propagation of magnetic skyrmions on elastically
deformable geometries by employing imaginary time quantum field theory methods.
We demonstrate that the Euclidean action of the problem carries information of
the elements of the surface space metric, and develop a description of the
skyrmion dynamics in terms of a set of collective coordinates. We reveal that
novel curvature-driven effects emerge in geometries with non-constant
curvature, which explicitly break the translational invariance of flat space.
In particular, for a skyrmion stabilized by a curvilinear defect, an inertia
term and a pinning potential are generated by the varying curvature, while both
of these terms vanish in the flat-space limit.
