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| Associative Cones and Integrable Systems |
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Citation： |
Chuu-Lian TERNG,Shengli KONG,Erxiao WANG.Associative Cones and Integrable Systems[J].Chinese Annals of Mathematics B,2006,27(2):153~168 |
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Net amount： 739 |
Authors： |
Chuu-Lian TERNG; Shengli KONG;Erxiao WANG |
Foundation： |
Partially supported by NSF grant DMS-0529756. |
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| Abstract： |
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 sphere $S^6$. It is known that a cone over
a surface $M$ in $\rmS^6$ is an associative submanifold of $\R^7$
if and only if $M$ is almost complex in $\rmS^6$. 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 $\rmG_2/\rmT^2$, and use this to explain some
of the known results. Moreover, the equation for
${\rmS}^1$-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 |
Classification： |
53, 22E |
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