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Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in this $S^6$?

There are certainly a bunch of holomorphic $S^2$'s coming from the inclusions $\mathbb R^3\to\mathbb R^7$ compatible with the cross product. If these $S^2$'s are cut out transversally (seems likely), then one can start gluing them together at intersection points to get lots of other examples. Are there any other natural constructions?

Note that this gluing construction of lots of $S^2$'s (which are all null-homologous) shows that the moduli space of (even connected) pseudo-holomorphic curves is very non-compact (glue together arbitrarily many spheres). Of course, this does not contradict Gromov's compactness theorem since this almost complex structure $S^6$ is not tamed by a symplectic form (since $H^2(S^6;\mathbb R)=0$).

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It might be immodest of me to mention my own work, but there is quite a lot known about the pseudo-holomorphic curves in $S^6$. For example, it is known that the pseudoholomorphic rational curves are all algebraic and have area a multiple of $4\pi$ (and, yes, there are many of them). (It's true that the moduli space is not compact; however, because the area is quantized, the rational curves of a given area, can, in fact, be compactified to yield a compact moduli space, even an algebraic one.) As another example, every compact Riemann surface occurs as a pseudoholomorphic curve (possibly ramified) in $S^6$. (A student of Kevin Corlette, some years ago, proved that you can actually get every Riemann surface as an unramified pseudoholomorphic curve in $S^6$.)

For example, see my paper Submanifolds and special structures on the octonians, Journal of Differential Geometry 17 (1982), 184–232. (I regret very much the misspelling 'octonians' that permeates that paper.)

Added references: (January 2020) Recently, I was asked some questions about pseudo-holomorphic curves in $S^6$, and that inspired me to look at some of the literature that has been generated in the nearly 40 years since I wrote the paper above. A fair amount has been worked out, and anyone who wants to know more about the subject would benefit from consulting a few of the following papers:

  • J. Bolton, L. Vrancken, and L. Woodward, On almost complex curves in the nearly Kähler $6$-sphere, Q. J. Math. Oxford (2) 45 (1994), 407–427.

  • H. Hashimoto, Deformations of super-minimal $J$-holomorphic curves of a $6$-dimensional sphere, Tokyo J. Math. 27 (2004), 285–298.

  • J. K. Martins, Superminimal surfaces in the $6$-sphere, Bull. Braz. Math. Soc. (N.S.) 44 (2013), 25–48.

  • L. Fernández, The space of almost complex $2$-spheres in the $6$-sphere, Trans. Amer. Math. Soc. 367 (2015), 2437–2458.

  • M. Dajczer and T. Vlachos, A representation for pseudoholomorphic surfaces in spheres, Proc. Amer. Math. Soc. 144 (2016), 3105–3113.

This is by no means a complete list of relevant articles, but these do cover a range of interesting developments of the results in my 1982 JDG paper.

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  • $\begingroup$ You were not the only one to spell it like this. Compounded with the fact that some people call them Cayley numbers it makes hunting for references a bit challenging. :) $\endgroup$ – Vít Tuček Sep 24 '15 at 22:04
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    $\begingroup$ @Vit or octaves... $\endgroup$ – David Roberts Sep 25 '15 at 0:10
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    $\begingroup$ Someone should ask Google to identify all of these terms in its searches... $\endgroup$ – Simon Rose Sep 25 '15 at 9:12
  • $\begingroup$ I think the term "octaves" should be encouraged, because that was Graves' name for them when he invented them. Reinventors should probably not get to change the name, since they didn't read the literature carefully enough to see that there was already a name. $\endgroup$ – Ben McKay Jan 25 at 16:00

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