Figure 1: (a) The transverse spiral $Mn$ magnetic order found in many multiferroic perovskite manganites. Here the $Mn$ spins rotate around an axis defined by the cross product $Si×Si+1$, while the magnetic propagation $Q$ is parallel to the vector joining these two spins $ri,i+1$. The direction of the ferroelectric polarization is given by the cross product $P=(Si×Si+1)×Q$. In the same figure the $O$ atoms (red circles) are coherently displaced from their paraelectric position by a distance $d$ (depicted as white circles) and are related to the Dzyaloshinskii vector by $Di,i+1∼λd×ri,i+1$ , where $λ$ is the spin-orbit coupling constant. (Adapted from [2].) (b) A depiction of a weak ferromagnetism generated from a DM interaction from a small canting of antiferromagnetic spins that are stacked along the $c$ axis. For clarity only one column of spins along the $c$ axis is shown. In this arrangement the Dzyaloshinskii vector changes sign between pairs of spins.