8
$\begingroup$

Could any one tell me about the orientability of riemann surfaces? well, Holomorphic maps between two open sets of complex plane preserves orientation of the plane,I mean conformal property of holomorphic map implies that the local angles are preserved and in particular right angles are preserved, therefore the local notions of "clockwise" and "anticlockwise" for small circles are preserved. can we transform this idea to abstract riemann surfaces? thank you for your comments

$\endgroup$
7
  • $\begingroup$ I think a Riemann manifold of dimension $n$ is orientable if you can define a continuous $n$-form on it which doesn't vanish anywhere. A comment, not an answer, because I'm not sure it's really true. $\endgroup$
    – celtschk
    Jul 23, 2012 at 18:06
  • 2
    $\begingroup$ @celtschk: Note that a "Riemannian manifold" is not the same thing as a "Riemann surface". $\endgroup$ Jul 23, 2012 at 18:51
  • $\begingroup$ @HenningMakholm: Ah, I didn't know that. $\endgroup$
    – celtschk
    Jul 23, 2012 at 19:37
  • $\begingroup$ @celtschk But it's indeed the case, for the reason you state, that complex manifolds are canonically orientable. $\endgroup$ Jul 23, 2012 at 19:39
  • 1
    $\begingroup$ @DylanMoreland: it is a general fact about $n$-dimensional manifolds that they are orientable iff they admit a nowhere-vanishing $n$-form. This has nothing to do with a Riemannian metric or complex structure, so I don't understand your comment? $\endgroup$ Jul 23, 2012 at 20:01

3 Answers 3

27
$\begingroup$

Riemann surfaces are one dimensional complex manifolds. It is in fact true that any complex $n$-manifold is orientable as a real manifold. One possible way of showing this, is to calculate the determinant of the jacobians of the transition functions.

Suppose you have an $n$-dimensional complex manifold $M$. That is, $M$ is a (first countable, Hausdorff) space such that every point $p\in M$ has an open neighborhood $V$ homeomorphic to an open subset $\Omega\subset\mathbb C^n$, $$\phi:V\rightarrow\Omega~\mathrm{homeomorphism}$$ and when two such neighborhoods intersect, the transition functions are holomorphic, $$\psi\circ\phi^{-1}:\phi(V\cap V')\rightarrow\psi(V\cap V')~\mathrm{biholomorphism.}$$ By identifying $\mathbb C^n$ with $\mathbb R^n$ like so: $(z^1,\dots,z^n)\leftrightarrow(x^1,y^1,\dots,x^n,y^n)$ where $z^k=x^k+iy^k$ is the real part/imaginary part decomposition, we get a real manifold structure on $M$. We can calculate the jacobian matrices of the transition functions for both structures.

Let's set up some notation. The first coordinate chart will be $\phi:V\rightarrow\Omega, p\mapsto (z^1(p),\dots,z^n(p))$, and the second will be $\psi:V'\rightarrow\Omega', p\mapsto (Z^1(p),\dots,Z^n(p)).$ In the complex case, we have $$\mathrm{Jac}_{\phi(p)}(\psi\circ\phi^{-1})=\left(\frac{\partial~[\psi\circ\phi^{-1}]^k}{\partial~z^l}\right)_{1\leq k,l\leq n}=\left(\frac{\partial~Z^k(z^1,\dots,z^n)}{\partial~z^l}\right)_{1\leq k,l\leq n}\\=\left(c_l^k\right)_{1\leq k,l\leq n}\in\mathrm{GL}(n,\mathbb C),$$ and in the real case you'll find $$\mathrm{Jac}_{\phi(p)}(\psi^{\mathbb R}\circ(\phi^{\mathbb R})^{-1})= \left(\begin{array}{cc} \frac{\partial~X^k(x^1,y^1,\dots,x^n,y^n)}{\partial~x^l} & \frac{\partial~X^k(x^1,y^1,\dots,x^n,y^n)}{\partial~y^l} \\ \frac{\partial~Y^k(x^1,y^1,\dots,x^n,y^n)}{\partial~x^l} & \frac{\partial~Y^k(x^1,y^1,\dots,x^n,y^n)}{\partial~y^l} \end{array}\right)_{1\leq k,l\leq n}= \left(\begin{array}{cc} \mathrm{Re}(c_l^k) & -\mathrm{Im}(c_l^k) \\ \mathrm{Im}(c_l^k) & \mathrm{Re}(c_l^k) \end{array}\right)_{1\leq k,l\leq n}\in\mathrm{GL}(2n,\mathbb R),$$ using the Cauchy-Riemann equations. We will calculate the determinant of these matrices, show that it is always $>0$, which is equivalent to $M^{\mathbb R}$ being orientable.

We move on to calculating the determinants of these. Consider the $\mathbb R$-algebra homomorphism $$\rho:M_n(\mathbb C)\rightarrow M_{2n}(\mathbb R),~\left(c_l^k\right)_{1\leq k,l\leq n}\mapsto \left(\begin{array}{cc} \mathrm{Re}(c_l^k) & -\mathrm{Im}(c_l^k) \\ \mathrm{Im}(c_l^k) & \mathrm{Re}(c_l^k) \end{array}\right)_{1\leq k,l\leq n}$$ Since it is $\mathbb R$-linear and the spaces involved are finite dimensional, it is continuous. Also, being an algebra homomorphism, we have $\mathrm{det}~\rho(P^{-1}AP)=\mathrm{det}(\rho(P)^{-1}\rho(A)\rho(P))=\mathrm{det}(\rho(A))$. Finally, the diagonalizable matrices are dense in $M_n(\mathbb C)$, so we can restrict our calculations to diagonal matrices in $M_n(\mathbb C)$. For these, the calculations are easy: $$\mathrm{det}\big(\rho(\mathrm{Diag}(c_1,\dots,c_n))\big)=\mathrm{det}~\mathrm{Diag}\left(\left(\begin{array}{cc} \mathrm{Re}(c_1) & -\mathrm{Im}(c_1) \\ \mathrm{Im}(c_1) & \mathrm{Re}(c_1) \end{array}\right),\dots,\left(\begin{array}{cc} \mathrm{Re}(c_n) & -\mathrm{Im}(c_n) \\ \mathrm{Im}(c_n) & \mathrm{Re}(c_n) \end{array}\right)\right)\\=\prod_{i=1}^n\mathrm{det}\left(\begin{array}{cc} \mathrm{Re}(c_i) & -\mathrm{Im}(c_i) \\ \mathrm{Im}(c_i) & \mathrm{Re}(c_i) \end{array}\right)=\prod_{i=1}^n|c_i|^2=\left|\mathrm{det}~\mathrm{Diag}(c_1,\dots,c_n)\right|^2,$$ so we conclude that $$\forall A\in M_n(\mathbb C),~\mathrm{det}~\rho(A)=|\mathrm{det}~A|^2$$

Finally, we can conclude that the transition functions for the charts $\phi^{\mathbb R}$ for $M^{\mathbb R}$ have positive determinants, thus the real underlying manifold $M^{\mathbb R}$ is orientable.

$\endgroup$
7
  • $\begingroup$ Very nice answer Olivier. To clarify a point: I have heard the argument "since the diagonalizable matrices are dense in $M_n(\mathbb{C})$ we can work with diagonalisable matrices" before but I have never seen how it actually works. Does it involve some kind of limiting procedure: you start with a sequence of diagonalisable matrices which converge to your (not diagonalisable) matrix, apply $\rho$ and then take the limit? $\endgroup$
    – GFR
    Jan 16, 2014 at 11:26
  • $\begingroup$ @GFR thanks! Yes: if $A$ is a complex matrix, there is a sequence $(A_n)$ of diagonalisable matrices converging to $A$. By continuity of the determinant, we have $$\det(A)=\lim_{n\to\infty}\det(A_n)$$ Actually, my argument is redundant if you know that every complex matrix is similar to an upper triangular matrix. This upper triangular matrix translates to an upper triangular real matrix of $2\times 2$ blocs, and the determinant of an upper triangular bloc matrix being the product of the determinants of the diagonal blocs gives the answer again. So no limits are required after all. $\endgroup$ Jan 16, 2014 at 12:48
  • $\begingroup$ @OlivierBégassat, yes, but if you work with the upper triangular bloc you'd have to prove that the 2x2 blocs have positive determinant right? How do you know what they are? $\endgroup$ Jun 24, 2017 at 20:43
  • 1
    $\begingroup$ @SeñorBilly They will be of the form $$\begin{pmatrix}a_k&-b_k\\b_k&a_k\end{pmatrix}$$ where $a_k+ib_k=\lambda_k$ are the real part and imaginary part of the diagonal coefficent $\lambda_k\in\Bbb C$. $\endgroup$ Jun 24, 2017 at 21:58
  • $\begingroup$ Oh! Because we first triangulate in $\mathbb{C}$ and then we apply your reasoing? $\endgroup$ Jun 24, 2017 at 22:03
8
$\begingroup$

Holomorphic maps not only preserve nonoriented angles but oriented angles as well. Note that the map $z\mapsto\bar z$ preserves nonoriented angles, in particular nonoriented right angles, but it does not preserve the clockwise or anticlockwise orientation of small circles.

Now a Riemann surface $S$ is covered by local coordinate patches $(U_\alpha, z_\alpha)_{\alpha\in I}\ $, and the local coordinate functions $z_\alpha$ are related in the intersections $U_\alpha\cap U_\beta$ by means of holomorphic transition functions $\phi_{\alpha\beta}$: One has $z_\alpha=\phi_{\alpha\beta}\circ z_\beta$ where the Jacobian $|\phi'_{\alpha\beta}(z)|^2$ is everywhere $>0$. It follows that the positive ("anticlockwise") orientation of infinitesimal circles on $S$ is well defined througout $S$. In all, a Riemann surface $S$ is oriented to begin with.

For better understandng consider a Moebius band $M:=\{z\in {\mathbb C}\ | \ -1<{\rm Im}\, z <1\}/\sim\ ,$ where points $x+iy$ and $x+1-iy$ $\ (x\in{\mathbb R})$ are identified. In this case it is not possible to choose coordinate patches $(U_\alpha, z_\alpha)$ covering all of $M$ such that the transition functions are properly holomorphic. As a consequence this $M$ is not a Riemann surface, even though nonoriented angles make sense on $M$.

$\endgroup$
3
  • $\begingroup$ Hi Professor Blatter, does the locally orientation-preserving property of holomorphic maps mean that a domain to the left of a counterclockwise-oriented curve, under a holomorphic mapping, will again be to the left of the image curve? I just want to be careful that "locally" orientation-preserving implies what I am trying to claim (not sure if this is called domain-preservation?). Thanks @ChristianBlatter, $\endgroup$
    – User001
    Nov 19, 2015 at 1:47
  • $\begingroup$ Or, is this more of a connected set / continuity argument? Thanks @ChristianBlatter $\endgroup$
    – User001
    Nov 19, 2015 at 2:26
  • 1
    $\begingroup$ @LebronJames: Yes. Look what the map $f: z\mapsto1/z$ defined on $\dot{\mathbb C}$ does to the unit circle $\partial D$ oriented counterclockwise and to the punctured disk $\dot D$ lying to the left of $\partial D$. As a set $f(\partial D)=\partial D$, but $f(\partial D)$ is oriented clockwise. On the other hand $f(\dot D)$ is the exterior of $D$, which lies to the left of $f(\partial D)$. $\endgroup$ Nov 19, 2015 at 9:09
2
$\begingroup$

Every coordinate-change of a holomorphic atlas is by definition a bi-holomorphism between opens of $\mathbb{C}^n$, i.e. its derivative (tangent map) is a $\mathbb{C}$-linear isomorphism of $\mathbb{R}^{2n}$, hence has positive determinant.

($n=1$ for a Riemann surface)

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .