View source for Double of a Riemann surface

A two-sheeted [[Covering surface|covering surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339201.png" /> of a finite [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339202.png" />. Each interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339203.png" /> is brought into correspondence with a pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339205.png" /> of the double <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339206.png" />; in other words, two conjugate points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339208.png" /> are situated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d0339209.png" />. Each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392010.png" /> of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392011.png" /> is brought into correspondence with a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392012.png" />. Moreover, two non-intersecting neighbourhoods of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392013.png" /> are situated over each neighbourhood of an interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392015.png" /> is a [[Local uniformizing parameter|local uniformizing parameter]] in a neighbourhood of the interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392016.png" />, it will also be a local uniformizing parameter in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392017.png" />-neighbourhood of one out of the two conjugate points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392018.png" /> lying over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392019.png" />, say in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392020.png" />-neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392021.png" />; then, in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392022.png" />-neighbourhood of the conjugate point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392023.png" />, the complex conjugate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392024.png" /> of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392025.png" /> will be a local uniformizing parameter. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392026.png" /> is a local uniformizing parameter at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392027.png" /> of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392028.png" />, then the variable which is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392029.png" /> on one sheet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392030.png" /> and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392031.png" /> on the other will be a local uniformizing parameter at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392032.png" /> lying over it.

In the case of a compact orientable Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392033.png" />, the double simply consists of two compact orientable Riemann surfaces, and its study is accordingly of no interest. In all other cases the double of the Riemann surface is a compact orientable Riemann surface. This fact permits one to simplify the study of certain problems in the theory of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392034.png" /> by reducing them to the study of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392035.png" />. The genus (cf. [[Genus of a surface|Genus of a surface]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392036.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392038.png" /> is the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392040.png" /> is the number of components of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392041.png" />, which are assumed to be non-degenerate. For instance, the double of a simply-connected plane domain is a sphere, while the double of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392042.png" />-connected plane domain is a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392043.png" /> handles.

Analytic differentials on a Riemann surface (cf. [[Differential on a Riemann surface|Differential on a Riemann surface]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392044.png" /> are analytic differentials on the double <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392045.png" /> characterized by the fact that they assume conjugate values at conjugate points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392046.png" /> and take real values at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392047.png" /> lying over points of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392048.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Schiffer,   D.C. Spencer,   "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Picard,   "Traité d'analyse" , '''2''' , Gauthier-Villars  (1926)</TD></TR></table>



====Comments====
The process of constructing the double of a Riemann surface is called duplication. This process can be applied to any connected [[Two-dimensional manifold|two-dimensional manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392049.png" /> with boundary (cf. [[Boundary (of a manifold)|Boundary (of a manifold)]]) to yield a regular imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033920/d03392050.png" /> in a connected two-dimensional manifold (cf. [[#References|[a1]]], § 13.H).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,   L. Sario,   "Riemann surfaces" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.M. Farkas,   I. Kra,   "Riemann surfaces" , Springer  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR></table>