Figure 1: (a) The transverse spiral $\text{Mn}$ magnetic order found in many multiferroic perovskite manganites. Here the $\text{Mn}$ spins rotate around an axis defined by the cross product ${\mathbf{\text{S}}}_i\times {\mathbf{\text{S}}}_{i+1}$, while the magnetic propagation $\mathbf{\text{Q}}$ is parallel to the vector joining these two spins ${\mathbf{\text{r}}}_{i,i+1}$. The direction of the ferroelectric polarization is given by the cross product $\mathbf{\text{P}}=({\mathbf{\text{S}}}_i\times {\mathbf{\text{S}}}_{i+1})\times \mathbf{\text{Q}}$. In the same figure the $\text{O}$ 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 ${\mathbf{\text{D}}}_{i,i+1}\sim \lambda d\times {\mathbf{\text{r}}}_{i,i+1}$ , where $\lambda$ 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.