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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.
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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.
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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" />.
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A two-sheeted [[Covering surface|covering surface]]  $  W $
 +
of a finite [[Riemann surface|Riemann surface]]  $  R $.
 +
Each interior point  $  p \in R $
 +
is brought into correspondence with a pair of points  $  p $
 +
and  $  \widetilde{p}  $
 +
of the double  $  W $;
 +
in other words, two conjugate points  $  p $
 +
and  $  \widetilde{p}  $
 +
are situated over  $  p $.
 +
Each point  $  q $
 +
of the boundary of  $  R $
 +
is brought into correspondence with a point  $  q \in W $.
 +
Moreover, two non-intersecting neighbourhoods of the points  $  p , \widetilde{p}  \in W $
 +
are situated over each neighbourhood of an interior point  $  p \in R $.
 +
If  $  z $
 +
is a [[Local uniformizing parameter|local uniformizing parameter]] in a neighbourhood of the interior point  $  p \in R $,
 +
it will also be a local uniformizing parameter in a  $  W $-
 +
neighbourhood of one out of the two conjugate points of  $  W $
 +
lying over  $  p $,
 +
say in a  $  W $-
 +
neighbourhood of the point  $  p \in W $;
 +
then, in a  $  W $-
 +
neighbourhood of the conjugate point  $  \widetilde{p}  $,
 +
the complex conjugate  $  \overline{z}\; $
 +
of the variable  $  z $
 +
will be a local uniformizing parameter. If  $  z $
 +
is a local uniformizing parameter at a point  $  q $
 +
of the boundary of  $  R $,
 +
then the variable which is equal to  $  z $
 +
on one sheet of  $  W $
 +
and to  $  \overline{z}\; $
 +
on the other will be a local uniformizing parameter at the point  $  q \in W $
 +
lying over it.
 +
 
 +
In the case of a compact orientable Riemann surface  $  R $,
 +
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  $  R $
 +
by reducing them to the study of functions on  $  W $.
 +
The genus (cf. [[Genus of a surface|Genus of a surface]]) of  $  W $
 +
is  $  g + m - 1 $,
 +
where  $  g $
 +
is the genus of  $  R $
 +
and  $  m $
 +
is the number of components of the boundary of  $  R $,
 +
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  $  m $-
 +
connected plane domain is a sphere with  $  m - 1 $
 +
handles.
 +
 
 +
Analytic differentials on a Riemann surface (cf. [[Differential on a Riemann surface|Differential on a Riemann surface]]) $  R $
 +
are analytic differentials on the double $  W $
 +
characterized by the fact that they assume conjugate values at conjugate points of $  W $
 +
and take real values at the points $  q \in W $
 +
lying over points of the boundary of $  R $.
  
 
====References====
 
====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>
 
<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====
 
====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).
+
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]] $  M $
 +
with boundary (cf. [[Boundary (of a manifold)|Boundary (of a manifold)]]) to yield a regular imbedding of $  M $
 +
in a connected two-dimensional manifold (cf. [[#References|[a1]]], § 13.H).
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:36, 5 June 2020


A two-sheeted covering surface $ W $ of a finite Riemann surface $ R $. Each interior point $ p \in R $ is brought into correspondence with a pair of points $ p $ and $ \widetilde{p} $ of the double $ W $; in other words, two conjugate points $ p $ and $ \widetilde{p} $ are situated over $ p $. Each point $ q $ of the boundary of $ R $ is brought into correspondence with a point $ q \in W $. Moreover, two non-intersecting neighbourhoods of the points $ p , \widetilde{p} \in W $ are situated over each neighbourhood of an interior point $ p \in R $. If $ z $ is a local uniformizing parameter in a neighbourhood of the interior point $ p \in R $, it will also be a local uniformizing parameter in a $ W $- neighbourhood of one out of the two conjugate points of $ W $ lying over $ p $, say in a $ W $- neighbourhood of the point $ p \in W $; then, in a $ W $- neighbourhood of the conjugate point $ \widetilde{p} $, the complex conjugate $ \overline{z}\; $ of the variable $ z $ will be a local uniformizing parameter. If $ z $ is a local uniformizing parameter at a point $ q $ of the boundary of $ R $, then the variable which is equal to $ z $ on one sheet of $ W $ and to $ \overline{z}\; $ on the other will be a local uniformizing parameter at the point $ q \in W $ lying over it.

In the case of a compact orientable Riemann surface $ R $, 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 $ R $ by reducing them to the study of functions on $ W $. The genus (cf. Genus of a surface) of $ W $ is $ g + m - 1 $, where $ g $ is the genus of $ R $ and $ m $ is the number of components of the boundary of $ R $, 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 $ m $- connected plane domain is a sphere with $ m - 1 $ handles.

Analytic differentials on a Riemann surface (cf. Differential on a Riemann surface) $ R $ are analytic differentials on the double $ W $ characterized by the fact that they assume conjugate values at conjugate points of $ W $ and take real values at the points $ q \in W $ lying over points of the boundary of $ R $.

References

[1] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)
[2] E. Picard, "Traité d'analyse" , 2 , Gauthier-Villars (1926)

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 $ M $ with boundary (cf. Boundary (of a manifold)) to yield a regular imbedding of $ M $ in a connected two-dimensional manifold (cf. [a1], § 13.H).

References

[a1] L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1974)
[a2] H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980)
[a3] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
How to Cite This Entry:
Double of a Riemann surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double_of_a_Riemann_surface&oldid=17837
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article