  The influence of the geometrical curvature of chiral magnetic films on the
static and dynamic properties of hosted skyrmions are studied theoretically. We
predict the effects of the curvature-induced drift of skyrmions under the
action of the curvature gradients without any external stimuli. The strength of
the curvature-induced driving force essentially depends on the skyrmion type,
Neel or Bloch, while the trajectory of motion is determined by the type of
magnetic ordering: ferro- or antiferromagnetic. When moving on the surface,
skyrmions undergo deformations that depend on the type of skyrmion. In the
small-curvature limit, using the collective-variable approach we show, that the
driving force acting on a Neel skyrmion is linear in the gradient of the mean
curvature. The driving acting on a Bloch skyrmion is much smaller: it is
proportional to the product of the mean curvature and its gradient. In contrast
to the fast Neel skyrmions, the dynamics of the slow Bloch skyrmions is
essentially affected by the skyrmion profile deformation. For the sake of
simplicity, we restrict ourselves to the case of zero Gaussian curvature and
consider cylindrical surfaces of general type. Equations of motion for
ferromagnetic and antiferromagnetic skyrmions in curved magnetic films are
obtained in terms of collective variables. All analytical predictions are
confirmed by numerical simulations.
