Revista de la
Unión Matemática Argentina
Pseudoholomorphic curves in $\mathbb{S}^6$ and $\mathbb{S}^5$
Jost-Hinrich Eschenburg and Theodoros Vlachos
Volume 60, no. 2 (2019), pp. 517–537

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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.