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

Abstract

In this paper, by using the $G_2$-structure on Im$(\mathbb O)\cong\mathbb R^7$ from the octonions $\mathbb O$, the $G_2$-binormal motion of curves $\gamma(t,s)$ in $\mathbb R^7$ associated to the almost complex structure on $\mathbb S^6$ 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 $\Sigma$ in Im$(\mathbb O)$ swept by $\gamma(t,s)$ are determined.