  Easy axis antiferromagnets are robust against external magnetic fields of
moderate strength. Spin reorientations in strong fields can provide an insight
into more subtle properties of antiferromagnetic materials, which are often
hidden by their high ground state symmetry. Here, we investigate theoretically
effects of curvature in ring-shaped antiferromagnetic achiral anisotropic spin
chains in strong magnetic fields. We identify the geometry-governed
helimagnetic phase transition above the spin-flop field between vortex and
onion states. The curvature-induced Dzyaloshinskii-Moriya interaction results
in the spin-flop transition being of first- or second-order depending on the
ring curvature. Spatial inhomogeneity of the N\'eel vector in the spin-flop
phase generates weak ferromagnetic response in the plane perpendicular to the
applied magnetic field. Our work contributes to the understanding of the
physics of curvilinear antiferromagnets in magnetic fields and guides
prospective experimental studies of geometrical effects relying on
spin-chain-based nanomagnets.
