Talk:PlanetPhysics/Construction of Riemann Surface Using Paths

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: construction of Riemann surface using paths %%% Primary Category Code: 02.30.Fn %%% Filename: ConstructionOfRiemannSurfaceUsingPaths.tex %%% Version: 1 %%% Owner: rspuzio %%% Author(s): rspuzio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Note: All arcs and curves are assumed to be smooth in this entry.

Let $f$ be a complex \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} defined in a disk $D$ about a point $z_0 \in \mathbb{C}$. In this entry, we shall show how to construct a \htmladdnormallink{Riemann surface}{http://planetphysics.us/encyclopedia/RiemannSurface.html} such that $f$ may be analytically continued to a function on this surface by considering paths in the complex plane.

Let ${\cal P}$ denote the class of paths on the complex plane having $z_0$ as an endpoint along which $f$ may be analytically continued. We may define an \htmladdnormallink{equivalence relation}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\sim$ on this set --- $C_1 \sim C_2$ if $C_1$ and $C_2$ have the same endpoint and there exists a one-parameter family of paths along which $f$ can be analytically continued which includes $C_1$ and $C_2$.

Define ${\cal S}$ as the quotient of ${\cal P}$ modulo $\sim$. It is possible to extend $f$ to a function on ${\cal S}$. If $C \in {\cal P}$, let $f(C)$ be the value of the analytic continuation of $f$ at the endpoint of $C$ (not $z_0$, of course, but the other endpoint). By the monodromy \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}, if $C_1 \sim C_2$, then $f(C_1) = f(C_2)$. Hence, $f$ is well defined on the quotient ${\cal S}$.

Also, note that there is a natural projection map $\pi \colon {\cal S} \to \mathbb{C}$. If $C$ is an equivalence class of paths in ${\cal S}$, define $\pi (C)$ to be the common endpoint of those paths (not $z_0$, of course, but the other endpoint).

Next, we shall define a class of subsets of ${\cal S}$. If $f$ can be analytically continued from along a path $C$ from $z_0$ to $z_1$ then there must exist an open disk $D'$ centered about $z_1$ in which the continuation of $f$ is analytic. Given any $z \in D'$, let $C(z)$ be the concatenation of the path $C$ from $z_0$ to $z_1$ and the straight line segment from $z_1$ to $z$ (which lies inside $D'$). Let $N(C, D') \subset {\cal S}$ be the set of all such paths.

We will define a topology of ${\cal S}$ by taking all these sets $N(C,D')$ as a basis. For this to be legitimate, it must be the case that, if $C_3$ lies in the intersection of two such sets, $N(C_1,D_1)$ and $N(C_2,D_2)$ there exists a basis element $N(C_3,D_3)$ contained in the intersection of $N(C_1,D_1)$ and $N(C_2,D_2)$. Since the endpoint of $C_3$ lies in the intersection of $D_1$ and $D_2$, there must exist a disk $D_3$ centered about this point which lies in the intersection of $D_1$ and $D_2$. It is easy to see that $N(C_3,D_3) \subset N(C_1,D_1) \cap N(C_2,D_2)$.

Note that this topology has the Hausdorff property. Suppose that $C_1$ and $C_2$ are distinct elements of ${\cal S}$. On the one hand, if $\pi (C_1) \ne \pi (C_2)$, then one can find disjoint open disks $D_1$ and $D_2$ centered about $C_1$ and $C_2$. Then $N(C_1,D_1) \cap N(C_2,D_2) = \emptyset$ because $\pi (N(C_1,D_1)) \cap \pi(N(C_2,D_2)) = D_1 \cap D_2 = \emptyset$. On the other hand, if $\pi (C_1) = \pi (C_2)$, then let $D_3$ be the smaller of the disks $D_1$ and $D_2$. Then $N(C_1,D_3) \cap N(C_2,D_3) = \emptyset$.

To complete the proof that ${\cal S}$ is a Riemann surface, we must exhibit coordinate neighborhoods and \htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. As coordinate neighborhoods, we shall take the neighborhoods $N(C,D)$ introduced above and as homomorphisms we shall take the restrictions of $\pi$ to these neighborhoods. By the way that these neighborhoods have been defined, every element of ${\cal S}$ lies in at least one such neighborhood. When the \htmladdnormallink{domains}{http://planetphysics.us/encyclopedia/Bijective.html} of two of these homomorphisms overlap, the \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} of one homomorphism with the inverse of the restriction of the other homomorphism to the overlap region is simply the \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} map in the overlap region, which is analytic. Hence, ${\cal S}$ is a Riemann surface.

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