$\mathbb S^6$上的殆复结构及相关的薛定谔流
The Almost Complex Structure on $\mathbb S^6$ and Related Schrodinger Flows
摘要：
本文利用八元数$\mathbb O$在Im $(\mathbb O)\cong\mathbb R^7$上的$G_2$ -结构，研究了与$\mathbb S^6$上的殆复结构相联系的$\mathbb R^7$中曲线$γ(t,s)$的$G_2$ -副法向运动.证明了该运动等价于从$\mathbb R^1$到$\mathbb S^6$的薛定谔流，也等价于一个三个未知复变函数的非线性薛定谔型系统，它推广了$\mathbb{R}^3$中曲线的副法向运动与聚焦非线性薛定谔方程之间的著名对应关系.确定了$γ(t,s)$扫过的Im $(\mathbb O)$中曲面$Σ$的一些相关几何性质。
英文摘要：
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 $γ(t,s)$ 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 $Σ$ in Im$(\mathbb O)$ swept by $γ(t,s)$ are determined.
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