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We study the nonlinear $\sigma$-model in an external magnetic field appliedon curved surfaces with rotational symmetry. The Euler-Lagrange equationsderived from the Hamiltonian yield the double sine-Gordon equation (DSG)provided the magnetic field is tuned with the curvature of the surface. A$2\pi$ skyrmion appears like a solution for this model and surface deformationsare predicted at the sector where the spins point in the opposite direction tothe magnetic field. We also study some specific examples by applying the modelon three rotationally symmetric surfaces: the cylinder, the catenoid and thehyperboloid. The coupling between a magnetic field and the curvature of thesubstract is an interesting result and we believe that this issue may berelevant to be applied in condensed matter systems, e.g., superconductors,nematic liquid crystals, graphene and topological insulators.

Coupling between magnetic field and curvature in Heisenberg spins on surfaces with rotational symmetry