  We study the geometrical conditions for stabilizing magnetic skyrmions in
cylindrical nanostrips and nanotubes of ferromagnetic materials with chiral
interactions. We obtain the low-temperature equilibrium state of the system
implementing a simulation annealing technique for a classical spin Hamiltonian
with competing isotropic exchange and chiral interactions, radial anisotropy
and an external field. We address the impact of surface curvature on the
formation, the shape and the size of magnetic skyrmions. We demonstrate that
the evolution of the skyrmion phase with the curvature is controlled by the
competition between two characteristic lengths, namely the curvature radius,
$R$ (geometrical length) and the skyrmion radius, $R_{sk}$ (physical length).
In narrow nanotubes ($R<R_{sk}$) the skyrmion phase evolves to a stripe phase,
while in wide nanotubes ($R>R_{sk}$) a mixed skyrmion-stripe phase emerges.
Most interestingly, the mixed phase is characterized by spatially separated
skyrmions from stripes owing to the direction of the applied field relative to
the surface normal. In the stability regime ($R \gtrsim R_{sk}$) skyrmions
remain circular and preserve their size as a consequence of their topological
protection. Zero-field skyrmions are shown to be stable on curved nanoelements
with free boundaries within the same stability region ($R\gtrsim R_{sk}$). The
experimental and technological perspectives from the stability of skyrmions on
cylindrical surfaces are discussed.
