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Pseudoholomorphic curves in $\mathbb{S}^6$ and $\mathbb{S}^5$
Volume 60, no. 2
(2019),
pp. 517–537
https://doi.org/10.33044/revuma.v60n2a16
Abstract
The octonionic cross product on $\mathbb{R}^7$ induces a nearly Kähler structure on
$\mathbb{S}^6$, the analogue of the Kähler structure of $\mathbb{S}^2$ given by the usual
(quaternionic) cross product on $\mathbb{R}^3$. Pseudoholomorphic curves with respect
to this structure are the analogue of meromorphic functions. They are
(super-)conformal minimal immersions. We reprove a theorem of Hashimoto
[Tokyo J. Math. 23 (2000), 137–159] giving an intrinsic characterization of
pseudoholomorphic curves in
$\mathbb{S}^6$ and (beyond Hashimoto's work) $\mathbb{S}^5$. Instead of the Maurer–Cartan
equations we use an embedding theorem into homogeneous spaces (here: $\mathbb{S}^6 =
G_2/SU_3$) involving the canonical connection.
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Published by the Unión Matemática Argentina |
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