| Associative Cones and Integrable Systems |
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| 引用本文： | Chuu-Lian TERNG. Associative Cones and Integrable Systems[J]. 数学年刊B辑(英文版), 2006, 27(2) |
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| 作者姓名： | Chuu-Lian TERNG |
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| 作者单位： | Dedicated to |
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| 基金项目： | Partially supported by NSF grant DMS-0529756. |
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| 摘 要： | 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 S6. 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 are 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, and a discussion of periodic solutions is given.
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Associative Cones and Integrable Systems |
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| Chuu-Lian TERNG,Shengli KONG,Erxiao WANG. Associative Cones and Integrable Systems[J]. Chinese Annals of Mathematics,Series B, 2006, 27(2) |
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| Authors: | Chuu-Lian TERNG Shengli KONG Erxiao WANG |
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| Affiliation: | Dedicated to the memory of Shiing-Shen Chern |
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| 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 S6. 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 are 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, and a discussion of periodic solutions is given. |
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| Keywords: | Octonions Associative cone Almost complex curve Primitive map |
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