4
$\begingroup$

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!

$\endgroup$
0
1
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ 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". $\endgroup$ – Jerry Feb 17 '19 at 5:09
  • $\begingroup$ 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. $\endgroup$ – Travis Willse Feb 17 '19 at 5:30
  • $\begingroup$ 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”) $\endgroup$ – Jerry Feb 17 '19 at 5:58
  • $\begingroup$ 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. $\endgroup$ – Travis Willse Feb 17 '19 at 17:29
  • $\begingroup$ 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 $M$. $\endgroup$ – Travis Willse Feb 17 '19 at 17:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.