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Is there an analog of the statement of "every 2d oriented surface is a complex manifold"?

I saw a theorem in Blair's book, that "every 3d contact metric manifold is a strongly pseudo convex CR manifold". And we also know from Lutz and Martinet that every compact 3d manifold has a contact structure (and therefore a contact metric structure), it seems the analog would be "every compact 3-dimensional manifold is a strongly pseudo convex CR manifold".

So my question is:

1) Is this the right analog?

2) Introductory/review references on the 3d analog of "pseudo-holomorphic curve in symplectic manifold" (for instance, references on how they are defined, basic properties, counting invariants analog to G-W invariants)?

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Here’s one possible answer, but perhaps not the one you want. For one thing, it is more the three-dimensional analogue of “every two-dimensional oriented surface is symplectic”.

Let $(M,\omega)$ be an oriented three-dimensional manifold and $\omega$ a nowhere vanishing 3-form defining the orientation. If $f,g,h \in C^\infty(M)$ are smooth functions, define their Nambu 3-bracket $\lbrace f,g,h\rbrace$ by $$ df \wedge dg \wedge dh = \lbrace f,g,h\rbrace \omega $$ The Nambu 3-bracket defines on $C^\infty(M)$ the structure of a 3-Lie algebra.

If in addition $M$ is compact and without boundary, then $C^\infty(M)$ becomes a metric 3-Lie algebra, with inner product $$ \left( f, g\right) = \int_M f g \omega $$

Details can be found in these lecture notes of mine. (Apologies for the self-promotion.)

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  • $\begingroup$ Thanks for replying and the algebra is quite interesting. But as you stated it's not an analog I'm looking for. Hope there will be some other suggestions. $\endgroup$ – Lelouch Apr 13 '14 at 3:39

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