Associative Cones and Integrable Systems

الأصل · EN

We identify R⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit 6-sphere. It is known that a cone over a surface M in S⁶ is an associative submanifold of R⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ is the equation for primitive maps associated to the 6-symmetric space G₂/T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice associated to G₂, and a discussion of periodic solutions is given.