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It is known that for a complex manifold, coordinates exist such that the complex structure takes the canonical form $$ J=\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array}\right] $$ in a local patch. These coordinates are known as local holomorphic coordinates. Is there an analogous concept for quaternionic manifolds? I.e., are there special coordinates whereby the quaternionic structure is defined by the canonical form

$$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right] $$ $$ J_2= \left[\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right] $$

$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right]? $$

Mtheorist
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Briefly, "no": The quaternionic analogues of the Cauchy-Riemann equations (I've heard them called Cauchy-Feuter equations, see also MO Does Feuter regularity imply derivability in all directions?) imply affineness, so unless the manifold admits quaternionic charts whose overlap maps are affine (e.g., tori) there's no intrinsic notion of "quaternionic coordinates". (It's not the coordinates themselves that determine "holomorphicity", incidentally, it's the overlap maps. This point is often misunderstood.)

In case it's of interest, there are quaternionic and hyper-Kähler manifolds. Hyper-Kähler manifolds seem likely to be the differential-geometric structure you're looking for.

Andrew D. Hwang
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  • Thank you for your answer. I am indeed interested in hyperkaehler manifolds; in particular, I really need to know what are the coordinates on $\Sigma \times T^2$ (where $\Sigma$ is a Riemann surface) whereby the quaternionic structure is defined by the canonical form given above. – Mtheorist Jul 08 '17 at 15:43
  • Andrew, if I understand correctly, what you are saying is that hyperkaehler manifolds always admit local quaternionic coordinates whereby the quaternionic structure takes the canonical form locally. But how does one explicitly identify what these coordinates are for a particular hyperkaehler manifold, e.g., $T^4$? – Mtheorist Jul 09 '17 at 06:50
  • It's not quite that a hyper-Kähler manifold admits quaternionic coordinates in the sense of your question, but it does admit three holomorphic structures obeying the algebraic identities of generators $i$, $j$, $k$ for the quaternions. <> A $4$-dimensional torus may be viewed as $\mathbf{H}$ modulo a lattice (e.g., the lattice of integer quaternions $a + bi + cj + dk$ with $a$, $b$, $c$, $d$ integers). Cartesian coordinates define quaternionic coordinates in the sense of your question because the overlap maps are quaternionic translations. – Andrew D. Hwang Jul 09 '17 at 12:06
  • Hi Andrew, to clarify, what you are saying is that NOT all hyperkaehler manifolds admit coordinates whereby the three holomorphic structures you mentioned take the form given in my question, am I right? If yes, then are there any hyperkaehler manifolds, besides tori, which DO admit such coordinates? – Mtheorist Aug 31 '17 at 12:04