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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2018
- DOI:
- 10.48550/arXiv.1810.07985
- arXiv:
- arXiv:1810.07985
- Bibcode:
- 2018arXiv181007985D
- Keywords:
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- Mathematics - Differential Geometry;
- 32Q60;
- 53C44;
- 53C14;
- 53A04
- E-Print:
- 20 pages, comments are welcome