  Chiral antiferromagnets are currently considered for broad range of
applications in spintronics, spin-orbitronics and magnonics. In contrast to the
established approach relying on materials screening, the anisotropic and chiral
responses of low-dimensional antifferromagnets can be tailored relying on the
geometrical curvature. Here, we consider an achiral, anisotropic
antiferromagnetic spin chain and demonstrate that these systems possess
geometry-driven effects stemming not only from the exchange interaction but
also from the anisotropy. Peculiarly, the anisotropy-driven effects are
complementary to the curvature effects stemming from the exchange interaction
and rather strong as they are linear in curvature. These effects are
responsible for the tilt of the equilibrium direction of vector order
parameters and the appearance of the homogeneous Dzyaloshinskii-Moriya
interaction. The latter is a source of the geometry-driven weak ferromagnetism
emerging in curvilinear antiferromagnetic spin chains. Our findings provide a
deeper fundamental insight into the physics of curvilinear antiferromagnets
beyond the $\sigma$-model and offer an additional degree of freedom in the
design of spintronic and magnonic devices.
