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

TL;DR

This paper explores the relationship between the almost complex structure on the 6-sphere and Schrödinger flows, using $G_2$-structures and octonions to connect curve motions in $\

Contribution

It introduces a novel connection between $G_2$-binormal motion of curves in $\

Findings

Equivalence of curve motion to Schrödinger flows on $\

Generalization of the nonlinear Schrödinger equation in this geometric context

Geometric properties of the swept surface $\

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.