# Why is the matrix representation of an almost complex structure like this?

Let $$M$$ be a Riemann surface, $$J$$ be an almost complex structure (i.e. a 1-1 tensor such that $$J^2=-I$$ and for any $$x \in M,v,w \in T_xM, \{v,Jv\}$$ is oriented). Consider a conformal coordinate at a point $$x \in M$$. Then the text (page 24) that I am reading says that in conformal coordinate, $$J=\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$$.

This doesn't make any sense to me for the following reason: I know that there can be more than one almost complex structure, and the matrix representation uniquely determines a linear operator. As we just start with an arbitrary $$J$$, it is not possible for us to obtain a matrix representation that takes the special form I mentioned above, unless there is only one almost complex structure on $$M$$.

My thought is that $$J$$ only takes a form $$S\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}S^{-1}$$, but the drawback is that I have no idea how to prove the theorem in the text if $$J$$ takes this form.

Thank you very much for answering this question!

In the context of Riemann surfaces, $$J$$ is not an arbitrary complex structure but rather a particular complex structure that, depending on your definition of Riemann surface, is either essentially built therein or is fast consequence thereof.

In the latter case, if we define, e.g., a Riemann surface to be a (connected) oriented real smooth manifold $$M$$ of dimension $$2$$ equipped with a conformal structure $$\bf c$$, we can define $$J$$ as follows: Fix any representative metric $$g \in {\bf c}$$. Then, for $$p \in M$$ and $$X \in T_p M$$, define $$JX \in T_p M$$ to be the unique vector for which

1. The oriented angle between $$X$$ and $$JX$$ is $$\frac{\pi}{2}$$ (equivalently, $$g(X, JX) = 0$$ and $$(X, JX)$$ is an oriented basis of $$T_p M$$), and
2. $$X$$ and $$JX$$ have the same length ($$g(X, X) = g(JX, JX)$$).

Both statements are invariant under a change of metric $$g \in {\bf c}$$, so $$J$$ is well-defined (a priori rough) bundle map $$TM \to TM$$. From this definition, we see that $$J$$ is fiberwise linear and that for any $$X \in T_p M$$ the matrix representation of $$J_p = J \vert_{T_p M}$$ with respect to the basis $$(X, JX)$$ is the familiar $$\phantom{(\ast)}\qquad [J_p] = \pmatrix{0&-1\\1&0} . \qquad (\ast)$$

In any (oriented, smooth) conformal coordinates $$(x, y)$$, the conformal class contains $$dx^2 + dy^2$$, in the induced basis $$(\partial_x, \partial_y)$$ at any point $$p$$ applying the above definition shows that $$J \partial_x = \partial_y)$$, so the matrix representation $$[J_p]$$ of the coordinate basis is just $$(\ast)$$ as claimed. Since the components of $$[J_p]$$ are smooth functions, $$J$$ is a smooth bundle endomorphism of $$TM$$.

Remark A similar procedure lets us recover $${\bf c}$$ and the orientation on $$M$$ from $$J$$. This equivalence of the conformal structure (plus an orientation) and the (almost) complex structure is a consequence of the isomorphism between the oriented conformal group $$CSO(2)$$ and the complex general linear group $$GL(1, \Bbb C) \cong \Bbb C^*$$ special to this dimension.

• In my context, a Riemann surface is a topological space which can be equipped with (possibly different) complex structures. So I think that $J$ can be determined by different complex structures and thus "arbitrary". – Jerry Feb 17 '19 at 5:09
• That's certainly not a standard definition of Riemann surface (and there are a few practical problems with it). Where did you find that definition? It's certainly not the one (Definition 0.2) given in the reference linked in the question. – Travis Willse Feb 17 '19 at 5:30
• In chapter 1 of the reference, the author talks about the space of almost complex structures $\mathcal{A}$. He established the fact that given a complex structure, we can get a corresponding almost complex structure. Therefore, I don’t think that the author fixed any complex structure on the topological space $M$. (Maybe I have misused the term “Riemann surface”) – Jerry Feb 17 '19 at 5:58
• At that point the author has just fixed an underlying oriented, $C^{\infty}$ $2$-manifold $M$ and, like you say, is considering the space $\mathcal A$ of almost complex structures (compatible with the orientation). See the paragraph before Definition 0.5. In any case, in order to talk about conformal coordinates one must already have fixed a complex structure (or oriented conformal structure) in the first place, and that's the case on p. 24, where the term occurs in the proof of Lemma 1.3.5; here the conformal structure is just the $J$ that appears in the hypotheses of the lemma. – Travis Willse Feb 17 '19 at 17:29
• In short, different choices of $J$ lead to different Riemann surfaces. The central problem of Teichmuller Theory in this setting is to understand what are all the Riemann surface structures an underlying manifold $M$ can have, where we consider two structures to be the same if there is a diffeomorphism (homotopic to the identity) that takes one to the other. The space of possibilities is the Teichmuller moduli space $\mathcal{T}(M)$ of $M$. – Travis Willse Feb 17 '19 at 17:36