Topological magnetic solitons on a paraboloidal shell

•Curvature-induced change in the width of a skyrmion on a paraboloid.•Presence of 2π-skyrmions due to the interaction with external fields.•Changes in the width of a skyrmion induced by magnetic fields.•Coupling between magnetic field and curvature.•Prediction of vortex repulsion due to a paraboloidal shell.

We study the influence of curvature on the exchange energy of skyrmions and vortices on a paraboloidal surface. It is shown that such structures appear as excitations of the Heisenberg model, presenting topological stability, unlike what happens on other simply-connected geometries such as pseudospheres. We also show that the skyrmion width depends on the geometrical parameters of the paraboloid. The presence of a magnetic field leads to the appearance of 2π-skyrmions, introducing a new characteristic length into the system. Regarding vortices, the geometrical parameters of the paraboloid play an important role in the exchange energy of this excitation.