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 Schrödinger flows from ${\mathbb{R}}^1$ to ${\mathbb{S}}^6$, and also to a nonlinear Schrödinger-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 Schrödinger equation. Some related geometric properties of the surface $\Sigma$ in Im$(\mathbb{O})$ swept by $\gamma (t,s)$ are determined.

20 pages, comments are welcome