Associative Cones and Integrable Systems

Details Associative Cones and Integrable Systems Associative Cones and Integrable Systems

2006-02-25

arxiv.org/abs/math/0602565v1

Mathematics

Differential Geometry

to appear in Chinese Annals of Mathematics (2006), a special issue in memory of S. S. Chern

Scientific paper

We identify R^7 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 S^6 is an associative submanifold of R^7 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 is 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 associated to G_2, and a discussion of periodic solutions is given.

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