Associative Cones and Integrable System

Abstract

We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S ⁶. It is known that a cone over a surface M in S ⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S ⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S ⁶ are the equation for primitive maps associated to the 6-symmetric space G ₂=T ², and use this to explain some of the known results. Moreover, the equation for S ¹-symmetric almost complex curves in S ⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given.

Keywords

Octonions / Associative cone / Almost complex curve / Primitive map / 53 / 22E

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