The Almost Complex Structure on ${\mathbb{S}}^6$ and Related Schrodinger Flows

In this paper, by using the $G_2$-structure on Im$(\mathbb O)\cong\mathbb R^7$fromtheoctonions$\mathbb O$,the$G_2$-binormalmotionofcurves$\gamma(t,s)$in$\mathbb R^7$associatedtothealmostcomplexstructureon$\mathbb S^6$ is studied. The motion is proved to be equivalent to Schr"odinger flows from ${\mathbb{R}}^1$ to ${\mathbb{S}}^6$, and also to a nonlinear Schr"odinger-type system in three unknown complex functions that generalizes the famous correspondence between the binormal motion of curves in ${\mathbb{R}}^3$ and the focusing nonlinear Schr"odinger equation. Some related geometric properties of the surface $\mathrm{\Sigma }$ in Im$(\mathbb{O})$ swept by $\gamma (t,s)$ are determined.