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''of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820401.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820402.png" />''
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A surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820403.png" /> such that the [[Complete analytic function|complete analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820404.png" />, which is, in general, multiple-valued, can be considered as a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820405.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820406.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820407.png" />.
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The concept of a Riemann surface arose in connection with the studies of algebraic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820408.png" /> defined by an algebraic equation
+
''of an analytic function  $  w = f( z) $
 +
of a complex variable  $  z $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r0820409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A surface  $  R $
 +
such that the [[Complete analytic function|complete analytic function]]  $  w = f( z) $,
 +
which is, in general, multiple-valued, can be considered as a single-valued analytic function  $  w = F( p) $
 +
of a point  $  p $
 +
on  $  R $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204011.png" />, are polynomials with constant coefficients, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204013.png" />. In the works of V. Puiseux (1850–1851) one discovers a clear understanding of multiple-valuedness, characteristic of these functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204014.png" />, when to each value of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204016.png" /> values of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204017.png" /> correspond. B. Riemann (1851–1857, see [[#References|[1]]]) was the first to show how for any algebraic function one can construct a surface on which this function can be considered as a single-valued rational function of the point. The obtained Riemann surface can be identified with the [[Algebraic curve|algebraic curve]] defined by equation (1). In general, a mutual penetration (sometimes more intensive, sometimes less intensive) of ideas and methods of the theory of functions of a complex variable on the one hand and of algebra and algebraic geometry on the other hand is characteristic of the whole period of further development of the theory of Riemann surfaces, associated with the names of F. Klein, H. Poincaré, P. Koebe, and others. The landmark of this development was the first edition of the book of H. Weyl [[#References|[18]]], in which the general concept of an abstract Riemann surface was formulated.
+
The concept of a Riemann surface arose in connection with the studies of algebraic functions $  w = f( z) $
 +
defined by an algebraic equation
  
Definition A: A connected topological [[Hausdorff space|Hausdorff space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204018.png" /> is called an abstract Riemann surface or, simply, a Riemann surface, if it admits a covering by open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204019.png" /> together with a [[Homeomorphism|homeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204020.png" /> corresponding to each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204022.png" /> is the unit disc in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204023.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204024.png" />; moreover, if a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204025.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204027.png" />, then the one-to-one correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204028.png" /> should be a [[Conformal mapping|conformal mapping]] of the first kind in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204029.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204030.png" /> is a univalent analytic function in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204031.png" />. In other words, an abstract Riemann surface is a two-dimensional complex-analytic manifold.
+
$$ \tag{1 }
 +
a _ {0} ( z) w  ^ {m} + a _ {1} ( z) w  ^ {m-} 1 + \dots + a _ {m} ( z)  = 0,
 +
$$
  
The definition of a Riemann surface with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204032.png" /> differs from definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204033.png" /> by the fact that together with the homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204034.png" />, homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204035.png" /> are admitted, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204036.png" /> is the unit upper half-disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204037.png" />; moreover, it is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204038.png" /> is not already a Riemann surface in the sense of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204039.png" />. The points of a Riemann surface with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204040.png" /> that have neighbourhoods homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204041.png" /> are called interior and the other points, that are mapped to the points of the segment
+
where  $  a _ {j} ( z) $,
 +
$  j = 0 \dots m $,
 +
are polynomials with constant coefficients,  $  a _ {0} ( z) \neq 0 $,  
 +
$  a _ {m} ( z) \neq 0 $.  
 +
In the works of V. Puiseux (1850–1851) one discovers a clear understanding of multiple-valuedness, characteristic of these functions  $  w = f( z) $,
 +
when to each value of the variable  $  z $,
 +
$  m $
 +
values of the variable  $  w $
 +
correspond. B. Riemann (1851–1857, see [[#References|[1]]]) was the first to show how for any algebraic function one can construct a surface on which this function can be considered as a single-valued rational function of the point. The obtained Riemann surface can be identified with the [[Algebraic curve|algebraic curve]] defined by equation (1). In general, a mutual penetration (sometimes more intensive, sometimes less intensive) of ideas and methods of the theory of functions of a complex variable on the one hand and of algebra and algebraic geometry on the other hand is characteristic of the whole period of further development of the theory of Riemann surfaces, associated with the names of F. Klein, H. Poincaré, P. Koebe, and others. The landmark of this development was the first edition of the book of H. Weyl [[#References|[18]]], in which the general concept of an abstract Riemann surface was formulated.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204042.png" /></td> </tr></table>
+
Definition A: A connected topological [[Hausdorff space|Hausdorff space]]  $  R $
 +
is called an abstract Riemann surface or, simply, a Riemann surface, if it admits a covering by open sets  $  U $
 +
together with a [[Homeomorphism|homeomorphism]]  $  \alpha :  U \rightarrow D $
 +
corresponding to each set  $  U $,
 +
where  $  D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
 +
is the unit disc in the complex  $  z $-
 +
plane  $  \mathbf C $;  
 +
moreover, if a point  $  p \in R $
 +
belongs to  $  U $
 +
and  $  U  ^  \prime  $,
 +
then the one-to-one correspondence  $  z  ^  \prime  = \alpha  ^  \prime  \alpha  ^ {-} 1 ( z) $
 +
should be a [[Conformal mapping|conformal mapping]] of the first kind in a neighbourhood of the point  $  \alpha ( p) \in D $,
 +
that is,  $  z  ^  \prime  = \alpha  ^  \prime  \alpha  ^ {-} 1 ( z) $
 +
is a univalent analytic function in a neighbourhood of the point  $  \alpha ( p) \in D $.  
 +
In other words, an abstract Riemann surface is a two-dimensional complex-analytic manifold.
  
form the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204043.png" />. The set of interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204044.png" /> (the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204045.png" />) is a Riemann surface in the sense of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204046.png" />. Thus, in the case of a Riemann surface with boundary, the boundary is usually considered to be a non-empty set.
+
The definition of a Riemann surface with boundary  $  \overline{R}\; $
 +
differs from definition  $  A $
 +
by the fact that together with the homeomorphisms  $  \alpha :  U \rightarrow D $,
 +
homeomorphisms  $  \alpha :  U \rightarrow D _ {0}  ^ {+} $
 +
are admitted, where  $  D _ {0}  ^ {+} = \{ {z \in \mathbf C } : {| z | < 1,  \mathop{\rm Im}  z \geq  0 } \} $
 +
is the unit upper half-disc in  $  \mathbf C $;
 +
moreover, it is usually assumed that  $  \overline{R}\; $
 +
is not already a Riemann surface in the sense of definition $  A $.  
 +
The points of a Riemann surface with boundary $  \overline{R}\; $
 +
that have neighbourhoods homeomorphic to  $  D $
 +
are called interior and the other points, that are mapped to the points of the segment
  
A Riemann surface (with boundary) is a triangulable and orientable manifold with a countable base and, hence, it is separable and metrizable. A compact Riemann surface (without boundary) is called a closed Riemann surface; the wider class of finite Riemann surfaces includes the closed Riemann surfaces and the compact Riemann surfaces with a boundary consisting of a finite number of connected components. Non-compact Riemann surfaces with boundary or without it are called open Riemann surfaces. In certain cases it is more convenient to admit in definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204047.png" /> not only conformal mappings of the first kind but also conformal mappings of the second kind. A Riemann surface with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204048.png" /> (or without it) obtained using such an approach is, generally speaking, not orientable any more, but under the assumption that it be finite it can be conformally imbedded in an orientable closed Riemann surface: the double of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204049.png" /> (see [[#References|[8]]], cf. [[Double of a Riemann surface|Double of a Riemann surface]]).
+
$$
 +
\{ {z = x + iy \in \mathbf C } : {- 1 < x < 1, y = 0 } \}
 +
,
 +
$$
  
Let an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204050.png" /> be given by one of its regular elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204051.png" />, i.e. by a pair consisting of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204052.png" /> and a power series
+
form the boundary  $  \partial  \overline{R}\; $.  
 +
The set of interior points of  $  \overline{R}\; $(
 +
the interior of  $  \overline{R}\; $)
 +
is a Riemann surface in the sense of definition  $  A $.  
 +
Thus, in the case of a Riemann surface with boundary, the boundary is usually considered to be a non-empty set.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204053.png" /></td> </tr></table>
+
A Riemann surface (with boundary) is a triangulable and orientable manifold with a countable base and, hence, it is separable and metrizable. A compact Riemann surface (without boundary) is called a closed Riemann surface; the wider class of finite Riemann surfaces includes the closed Riemann surfaces and the compact Riemann surfaces with a boundary consisting of a finite number of connected components. Non-compact Riemann surfaces with boundary or without it are called open Riemann surfaces. In certain cases it is more convenient to admit in definition  $  A $
 +
not only conformal mappings of the first kind but also conformal mappings of the second kind. A Riemann surface with boundary  $  \overline{R}\; $(
 +
or without it) obtained using such an approach is, generally speaking, not orientable any more, but under the assumption that it be finite it can be conformally imbedded in an orientable closed Riemann surface: the double of  $  \overline{R}\; $(
 +
see [[#References|[8]]], cf. [[Double of a Riemann surface|Double of a Riemann surface]]).
  
with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204054.png" /> and radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204056.png" />. [[Analytic continuation|Analytic continuation]] of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204057.png" /> along all possible paths in the extended plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204058.png" /> allows one to obtain all regular elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204059.png" /> of the same type; in their totality they form the complete analytic function, which is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204060.png" />. Moreover, under analytic continuation elements of a more general nature arise:
+
Let an analytic function  $  w = f( z) $
 +
be given by one of its regular elements $  ( a, P) = ( a, P( z- a)) $,
 +
i.e. by a pair consisting of a point  $  a \in \mathbf C $
 +
and a power series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204061.png" /></td> </tr></table>
+
$$
 +
P( z- a)  = \sum _ {\nu = 0 } ^  \infty  a _  \nu  ( z- a)  ^  \nu
 +
$$
  
i.e. pairs consisting of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204062.png" /> and a generalized power series (a Puiseux series):
+
with centre  $  a $
 +
and radius of convergence  $  r( a) $,
 +
$  0 < r( a) \leq  \infty $.  
 +
[[Analytic continuation|Analytic continuation]] of the element  $  ( a, P) $
 +
along all possible paths in the extended plane  $  \overline{\mathbf C}\; $
 +
allows one to obtain all regular elements  $  ( b, Q) $
 +
of the same type; in their totality they form the complete analytic function, which is also denoted by  $  w = f( z) $.
 +
Moreover, under analytic continuation elements of a more general nature arise:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204063.png" /></td> </tr></table>
+
$$
 +
( b, S)  = ( b, S(( z- b)  ^ {1/n} )),
 +
$$
  
or (in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204064.png" /> is the point at infinity):
+
i.e. pairs consisting of a point $  b \in \overline{\mathbf C}\; $
 +
and a generalized power series (a Puiseux series):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204065.png" /></td> </tr></table>
+
$$
 +
S(( z- b)  ^ {1/n} )  = \sum _ {\nu = m } ^  \infty  b _ {n} ( z- b) ^ {\nu /n }
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204066.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204067.png" /> is a positive integer. Moreover, these series converge when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204068.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204069.png" />, respectively. The generalized elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204070.png" /> or, more precisely, their equivalence classes, form in their totality the [[Analytic image|analytic image]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204071.png" /> corresponding to the given analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204072.png" />. Among the equivalence classes of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204073.png" /> that form the analytic image one can distinguish regular ones, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204074.png" />, and ramified ones, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204075.png" />. The introduction of an appropriate topology on the analytic image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204076.png" /> will turn it into the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204077.png" /> of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204078.png" />. This can be achieved, for example, by defining the neighbourhood of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204080.png" />, as the set consisting of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204081.png" /> itself and all the regular elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204082.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204083.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204085.png" />, and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204086.png" /> converges to one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204087.png" /> determinations of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204088.png" /> in their common domain of definition, i.e.
+
or (in the case when $  b = \infty $
 +
is the point at infinity):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204089.png" /></td> </tr></table>
+
$$
 +
S( z  ^ {-} 1/n )  = \sum _ {\nu = m } ^  \infty  b _  \nu  z ^ {- \nu /n } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204090.png" /> is one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204091.png" />-th order roots of unity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204092.png" />. The neighbourhood of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204093.png" /> consists of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204094.png" /> itself and all those regular elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204095.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204096.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204098.png" />, and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r08204099.png" /> converges to one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040100.png" /> determinations of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040101.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040102.png" /> satisfies all the conditions of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040103.png" />.
+
where $  m $
 +
is an integer and  $  n $
 +
is a positive integer. Moreover, these series converge when  $  | z- b | < r( b) $
 +
or  $  | z | > r( \infty ) > 0 $,
 +
respectively. The generalized elements  $  ( b, S) $
 +
or, more precisely, their equivalence classes, form in their totality the [[Analytic image|analytic image]]  $  A _ {f} $
 +
corresponding to the given analytic function  $  w = f( z) $.  
 +
Among the equivalence classes of elements  $  ( b, S) $
 +
that form the analytic image one can distinguish regular ones, when  $  n= 1 $,
 +
and ramified ones, when  $  n> 1 $.  
 +
The introduction of an appropriate topology on the analytic image  $  A _ {f} $
 +
will turn it into the Riemann surface  $  R _ {f} $
 +
of the analytic function  $  w = f( z) $.  
 +
This can be achieved, for example, by defining the neighbourhood of an element  $  ( b, S) $,
 +
$  b \neq \infty $,
 +
as the set consisting of the element $  ( b, S) $
 +
itself and all the regular elements $  ( a, P) $
 +
of $  A _ {f} $
 +
for which $  | b- a | < \rho  ^ {n} $,  
 +
$  \rho < r( b) $,  
 +
and the series $  P( z- a) $
 +
converges to one of the $  n $
 +
determinations of the series $  S(( z- b)  ^ {1/n} ) $
 +
in their common domain of definition, i.e.
  
Thus, to any analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040104.png" /> corresponds a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040105.png" /> on which this function is a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040106.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040107.png" />. This means that in a neighbourhood of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040108.png" /> there exists a [[Local uniformizing parameter|local uniformizing parameter]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040109.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040110.png" /> is represented as a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040111.png" />. In other words, the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040112.png" /> of an analytic function is a geometric construct that is used for the global [[Uniformization|uniformization]] of a, generally speaking, multiple-valued relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040113.png" />. In a neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040114.png" /> this relation is uniformized by the two single-valued analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040116.png" />. On the other hand, the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040117.png" /> that takes each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040118.png" /> to its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040119.png" /> shows that the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040120.png" /> of an analytic function is a (ramified) [[Covering surface|covering surface]] over the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040121.png" /> or, which is the same, over the [[Riemann sphere|Riemann sphere]]. The projections of the ramified elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040122.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040123.png" /> are branch points of this covering.
+
$$
 +
P( z- a)  \equiv  S( \epsilon ( z- b)  ^ {1/n} ),
 +
$$
  
At the same time, to each a priori given Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040124.png" /> correspond infinitely-many analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040125.png" /> with precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040126.png" /> as Riemann surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040127.png" />. For the case of closed Riemann surfaces this statement was formulated and proved already by Riemann in 1851. The central point of the corresponding proof is the construction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040128.png" /> of harmonic functions with given singularities. The proof given by Riemann was based on an uncritical application of the so-called [[Dirichlet principle|Dirichlet principle]]; Koebe (1909) was the first to give a rigorous proof; later there appeared simpler proofs of this fundamental statement, among them those based on the properly applied Dirichlet principle (see, e.g. [[#References|[3]]], [[#References|[4]]], [[#References|[17]]], [[#References|[18]]]).
+
where  $  \epsilon $
 +
is one of the $  n $-
 +
th order roots of unity, $  \epsilon  ^ {n} = 1 $.  
 +
The neighbourhood of the element  $  ( \infty , S) $
 +
consists of the element  $  ( \infty , S) $
 +
itself and all those regular elements  $  ( a, P) $
 +
of  $  A _ {f} $
 +
for which  $  | a | > \rho  ^ {-} 1 $,  
 +
$  \rho < r( b) $,  
 +
and the series  $  P( z- a) $
 +
converges to one of the  $  n $
 +
determinations of the series  $  S( z  ^ {-} 1/n ) $.
 +
The space  $  R _ {f} $
 +
satisfies all the conditions of definition  $  A $.
  
Whatever the orientable topological surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040129.png" /> may be, one can construct a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040130.png" /> homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040131.png" />, i.e. a Riemann surface of the same topological type as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040132.png" />. Closed Riemann surfaces are topologically completely determined by one number — the genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040134.png" /> (cf. [[Genus of a surface|Genus of a surface]]). The topological type of such a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040135.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040136.png" /> is a sphere, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040137.png" /> — a torus, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040138.png" /> — a generalized torus, or a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040139.png" /> handles. By cutting a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040140.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040141.png" /> along some arc one obtains a digon with the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040142.png" /> as its topological model or normal form, indicating that the points of the sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040143.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040144.png" /> are identified; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040145.png" /> one has to carry out <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040146.png" /> [[Canonical sections|canonical sections]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040147.png" />, after which one obtains the normal form of a closed Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040148.png" /> — a polygon with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040149.png" /> pairwise-identified sides; the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040150.png" /> should indicate the order of appearance of the sides. For instance, in Fig. athe normal forms of a sphere for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040151.png" /> and a torus for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040152.png" /> are shown together with their symbols.
+
Thus, to any analytic function  $  w = f( z) $
 +
corresponds a Riemann surface $  R _ {f} $
 +
on which this function is a single-valued analytic function  $  w = F( p) $
 +
of a point  $  p = ( b, S) \in R _ {f} $.
 +
This means that in a neighbourhood of any point  $  p _ {0} = ( b, S) $
 +
there exists a [[Local uniformizing parameter|local uniformizing parameter]]  $  t = ( z- b)  ^ {1/n} $
 +
in which  $  w $
 +
is represented as a single-valued analytic function  $  w = P( t) = F( p) $.  
 +
In other words, the Riemann surface  $  R _ {f} $
 +
of an analytic function is a geometric construct that is used for the global [[Uniformization|uniformization]] of a, generally speaking, multiple-valued relation  $  w = f( z) $.  
 +
In a neighbourhood of each point  $  p _ {0} = ( b, S) \in R _ {f} $
 +
this relation is uniformized by the two single-valued analytic functions  $  z = b+ t  ^ {n} $
 +
and  $  w = S( t) $.
 +
On the other hand, the projection  $  \pi :  ( b, S) \rightarrow b $
 +
that takes each element  $  p _ {0} = ( b, S) \in R _ {f} $
 +
to its centre  $  b $
 +
shows that the Riemann surface  $  R _ {f} $
 +
of an analytic function is a (ramified) [[Covering surface|covering surface]] over the extended complex plane  $  \overline{\mathbf C}\; $
 +
or, which is the same, over the [[Riemann sphere|Riemann sphere]]. The projections of the ramified elements  $  ( b, S) $
 +
with  $  n > 1 $
 +
are branch points of this covering.
 +
 
 +
At the same time, to each a priori given Riemann surface $  R $
 +
correspond infinitely-many analytic functions  $  w = f( z) $
 +
with precisely  $  R $
 +
as Riemann surface,  $  R _ {f} = R $.  
 +
For the case of closed Riemann surfaces this statement was formulated and proved already by Riemann in 1851. The central point of the corresponding proof is the construction on  $  R $
 +
of harmonic functions with given singularities. The proof given by Riemann was based on an uncritical application of the so-called [[Dirichlet principle|Dirichlet principle]]; Koebe (1909) was the first to give a rigorous proof; later there appeared simpler proofs of this fundamental statement, among them those based on the properly applied Dirichlet principle (see, e.g. [[#References|[3]]], [[#References|[4]]], [[#References|[17]]], [[#References|[18]]]).
 +
 
 +
Whatever the orientable topological surface  $  S $
 +
may be, one can construct a Riemann surface  $  R $
 +
homeomorphic to  $  S $,  
 +
i.e. a Riemann surface of the same topological type as $  S $.  
 +
Closed Riemann surfaces are topologically completely determined by one number — the genus $  g $,
 +
0 \leq  g < + \infty $(
 +
cf. [[Genus of a surface|Genus of a surface]]). The topological type of such a Riemann surface $  R $
 +
for $  g = 0 $
 +
is a sphere, for $  g = 1 $—  
 +
a torus, for $  g > 1 $—  
 +
a generalized torus, or a sphere with $  g $
 +
handles. By cutting a Riemann surface $  R $
 +
of genus $  g = 0 $
 +
along some arc one obtains a digon with the symbol $  s = aa  ^ {-} 1 $
 +
as its topological model or normal form, indicating that the points of the sides $  a $
 +
and $  a  ^ {-} 1 $
 +
are identified; when $  g \geq  1 $
 +
one has to carry out $  2g $[[
 +
Canonical sections|canonical sections]] $  a _ {1} , b _ {1} \dots a _ {g} , b _ {g} $,  
 +
after which one obtains the normal form of a closed Riemann surface $  R $—  
 +
a polygon with $  4g $
 +
pairwise-identified sides; the symbol $  s = a _ {1} \dots $
 +
should indicate the order of appearance of the sides. For instance, in Fig. athe normal forms of a sphere for $  g= 0 $
 +
and a torus for $  g= 1 $
 +
are shown together with their symbols.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082040a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082040a.gif" />
Line 51: Line 220:
 
Figure: r082040a
 
Figure: r082040a
  
From the analytic point of view a closed Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040153.png" /> is characterized by the fact that it is the Riemann surface of some algebraic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040154.png" /> defined by an algebraic equation (1) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040155.png" />. This Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040156.png" /> can be imagined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040157.png" /> sheets extending over the Riemann sphere and mutually connected in a certain manner at the branch points and along some lines connecting these points (the manner of connecting is determined by the specific form of equation (1)). The genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040158.png" /> of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040159.png" /> can in this case be expressed as a function of the number of sheets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040160.png" /> and the orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040161.png" /> of the branch points by the [[Riemann–Hurwitz formula|Riemann–Hurwitz formula]]
+
From the analytic point of view a closed Riemann surface $  R $
 +
is characterized by the fact that it is the Riemann surface of some algebraic function $  w = f( z) $
 +
defined by an algebraic equation (1) of degree $  m $.  
 +
This Riemann surface $  R $
 +
can be imagined as $  m $
 +
sheets extending over the Riemann sphere and mutually connected in a certain manner at the branch points and along some lines connecting these points (the manner of connecting is determined by the specific form of equation (1)). The genus $  g $
 +
of the Riemann surface $  R $
 +
can in this case be expressed as a function of the number of sheets $  m $
 +
and the orders $  k _ {1} \dots k _ {s} $
 +
of the branch points by the [[Riemann–Hurwitz formula|Riemann–Hurwitz formula]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040162.png" /></td> </tr></table>
+
$$
 +
= \sum _ {\nu = 1 } ^ { s } 
 +
\frac{k _  \nu  - 1 }{2}
 +
- m + 1.
 +
$$
  
Finite Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040163.png" /> are topologically completely characterized by the genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040165.png" />, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040166.png" /> of connected components of the boundary; their topological type is a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040167.png" /> handles and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040168.png" /> holes. In the normal form of a finite Riemann surface, the number of sides is not necessarily even, some sides corresponding to components of the boundary that remain free are not identified. The notion of the genus can also be generalized to open Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040169.png" />, for example by exhausting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040170.png" /> by a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040171.png" /> of compact Riemann surfaces with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040172.png" />, belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040173.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040174.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040176.png" />. The genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040177.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040178.png" /> is set equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040179.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040180.png" /> is the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040181.png" />. This limit exists and is independent of the choice of the exhaustion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040182.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040183.png" />. However, the genus does not completely define the topological type of an open Riemann surface; the topological types of open Riemann surfaces can be rather diverse. Thus, in Fig.2a, Fig.2b two models with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040184.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040185.png" />, respectively, are shown.
+
Finite Riemann surfaces $  \overline{R}\; $
 +
are topologically completely characterized by the genus $  g $,
 +
0 \leq  g < \infty $,  
 +
and the number $  l $
 +
of connected components of the boundary; their topological type is a sphere with $  g $
 +
handles and $  l $
 +
holes. In the normal form of a finite Riemann surface, the number of sides is not necessarily even, some sides corresponding to components of the boundary that remain free are not identified. The notion of the genus can also be generalized to open Riemann surfaces $  R $,  
 +
for example by exhausting $  R $
 +
by a sequence $  \{ \overline{R}\; _  \nu  \} _ {\nu = 1 }  ^  \infty  $
 +
of compact Riemann surfaces with boundary $  \overline{R}\; _  \nu  $,  
 +
belonging to $  R $,  
 +
and such that $  \overline{R}\; _  \nu  $
 +
is contained in $  \overline{R}\; _ {\nu + 1 }  $,  
 +
$  \cup _ {\nu = 1 }  ^  \infty  \overline{R}\; _  \nu  = R $.  
 +
The genus $  g $
 +
of $  R $
 +
is set equal to $  g = \lim\limits _ {\nu \rightarrow \infty }  g _  \nu  $,  
 +
where $  g _  \nu  $
 +
is the genus of $  \overline{R}\; _  \nu  $.  
 +
This limit exists and is independent of the choice of the exhaustion $  \{ \overline{R}\; _  \nu  \} $,  
 +
0 \leq  g \leq  + \infty $.  
 +
However, the genus does not completely define the topological type of an open Riemann surface; the topological types of open Riemann surfaces can be rather diverse. Thus, in Fig.2a, Fig.2b two models with $  g= 0 $
 +
and $  g= 2 $,  
 +
respectively, are shown.
  
An important topological characteristic of a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040186.png" /> is the order of connectivity: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040187.png" /> is called simply connected if any simple closed curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040188.png" /> can be deformed continuously into a point without getting out of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040189.png" />, i.e. in other words, if the [[Fundamental group|fundamental group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040190.png" /> is trivial. Otherwise the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040191.png" /> is called multiply connected. The schlichtartig Riemann surfaces form an important class of Riemann surfaces; they are the Riemann surfaces (with boundary or without) that are split by any simple closed curve into non-intersecting parts. For example, in Fig.2a a topological model of a multiply-connected schlichtartig Riemann surface is shown. A schlichtartig Riemann surface necessarily has genus zero. A schlichtartig Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040192.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040194.png" />-connected if the minimum number of sections necessary to convert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040195.png" /> to a simply-connected Riemann surface is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040197.png" /> (see Fig.2b).
+
An important topological characteristic of a Riemann surface $  R $
 +
is the order of connectivity: $  R $
 +
is called simply connected if any simple closed curve in $  R $
 +
can be deformed continuously into a point without getting out of $  R $,  
 +
i.e. in other words, if the [[Fundamental group|fundamental group]] of $  R $
 +
is trivial. Otherwise the Riemann surface $  R $
 +
is called multiply connected. The schlichtartig Riemann surfaces form an important class of Riemann surfaces; they are the Riemann surfaces (with boundary or without) that are split by any simple closed curve into non-intersecting parts. For example, in Fig.2a a topological model of a multiply-connected schlichtartig Riemann surface is shown. A schlichtartig Riemann surface necessarily has genus zero. A schlichtartig Riemann surface $  R $
 +
is called $  n $-
 +
connected if the minimum number of sections necessary to convert $  R $
 +
to a simply-connected Riemann surface is equal to $  n- 1 $,  
 +
$  n \geq  1 $(
 +
see Fig.2b).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082040b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082040b.gif" />
Line 67: Line 283:
 
Figure: r082040c
 
Figure: r082040c
  
The topological properties of a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040198.png" /> do not completely characterize the analytic properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040199.png" />, i.e. the topological properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040200.png" /> do not completely characterize the behaviour of functions of different classes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040201.png" />. In particular, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040202.png" /> be a function on the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040203.png" /> with values on another Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040204.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040205.png" /> is called analytic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040206.png" /> if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040208.png" />, one can find local uniformizing parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040209.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040210.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040211.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040212.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040213.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040214.png" />, respectively, such that the composite function
+
The topological properties of a Riemann surface $  R $
 +
do not completely characterize the analytic properties of $  R $,  
 +
i.e. the topological properties of $  R $
 +
do not completely characterize the behaviour of functions of different classes on $  R $.  
 +
In particular, let $  f: R _ {1} \rightarrow R _ {2} $
 +
be a function on the Riemann surface $  R _ {1} $
 +
with values on another Riemann surface $  R _ {2} $.  
 +
The function $  f $
 +
is called analytic on $  R _ {1} $
 +
if for any point $  p _ {0} \in R _ {1} $,  
 +
$  f( p _ {0} ) = q _ {0} $,  
 +
one can find local uniformizing parameters $  t = \phi ( p) $
 +
in a neighbourhood of $  p _ {0} $
 +
on $  R _ {1} $
 +
and $  \tau = \psi ( q) $
 +
in a neighbourhood of $  q _ {0} $
 +
on $  R _ {2} $,  
 +
respectively, such that the composite function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040215.png" /></td> </tr></table>
+
$$
 +
\tau  = \psi \{ f[ \phi  ^ {-} 1 ( t)] \}  = g( t)
 +
$$
  
is an analytic function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040216.png" /> in a neighbourhood of the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040217.png" />. Two Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040218.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040219.png" /> are called conformally equivalent, or are said to belong to the same conformal class (cf. [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]]), if there exists an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040220.png" /> that gives a one-to-one mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040221.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040222.png" />. From the point of view of the behaviour of analytic functions on Riemann surfaces, conformally-equivalent Riemann surfaces are considered as one and the same Riemann surface, but topologically-equivalent Riemann surfaces are not always conformally equivalent.
+
is an analytic function of the complex variable $  t $
 +
in a neighbourhood of the value $  t _ {0} = \phi ( p _ {0} ) $.  
 +
Two Riemann surfaces $  R _ {1} $
 +
and $  R _ {2} $
 +
are called conformally equivalent, or are said to belong to the same conformal class (cf. [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]]), if there exists an analytic function $  f: R _ {1} \rightarrow R _ {2} $
 +
that gives a one-to-one mapping from $  R _ {1} $
 +
onto $  R _ {2} $.  
 +
From the point of view of the behaviour of analytic functions on Riemann surfaces, conformally-equivalent Riemann surfaces are considered as one and the same Riemann surface, but topologically-equivalent Riemann surfaces are not always conformally equivalent.
  
In terms of Riemann surfaces, the Riemann mapping theorem can be formulated as follows: Any simply-connected Riemann surface is conformally equivalent to one of the following three domains: 1) the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040223.png" />, i.e. the Riemann sphere (the elliptic case); 2) the finite complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040224.png" />, i.e. the punctured Riemann sphere (the parabolic case); or 3) the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040225.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040226.png" />, i.e. the Riemann sphere with a section of positive length (the hyperbolic case). An important result is that any schlichtartig Riemann surface is conformally equivalent to some canonical domain in the extended complex plane. As such a canonical domain one may take the entire extended plane with a finite or infinite number of sections parallel to the real axis; moreover, some of these sections may degenerate into points. As mentioned above, in the case of a simply-connected Riemann surface the canonical domain either has no sections (elliptic type), or the section degenerates into a point (parabolic type), or the section has a positive length (hyperbolic type). All three types of simply-connected Riemann surfaces are conformally different, although the last two of them are topologically equivalent. The problem of types, which has not yet (1991) been solved completely, consists in finding additional conditions under which a simply-connected Riemann surface will be of hyperbolic or parabolic type (see [[#References|[6]]], [[#References|[7]]], [[#References|[10]]], [[#References|[11]]], and [[Riemann surfaces, classification of|Riemann surfaces, classification of]]).
+
In terms of Riemann surfaces, the Riemann mapping theorem can be formulated as follows: Any simply-connected Riemann surface is conformally equivalent to one of the following three domains: 1) the extended complex plane $  \overline{\mathbf C}\; = \mathbf C \cup \{ \infty \} $,  
 +
i.e. the Riemann sphere (the elliptic case); 2) the finite complex plane $  \mathbf C $,  
 +
i.e. the punctured Riemann sphere (the parabolic case); or 3) the unit disc $  D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
 +
in $  \mathbf C $,  
 +
i.e. the Riemann sphere with a section of positive length (the hyperbolic case). An important result is that any schlichtartig Riemann surface is conformally equivalent to some canonical domain in the extended complex plane. As such a canonical domain one may take the entire extended plane with a finite or infinite number of sections parallel to the real axis; moreover, some of these sections may degenerate into points. As mentioned above, in the case of a simply-connected Riemann surface the canonical domain either has no sections (elliptic type), or the section degenerates into a point (parabolic type), or the section has a positive length (hyperbolic type). All three types of simply-connected Riemann surfaces are conformally different, although the last two of them are topologically equivalent. The problem of types, which has not yet (1991) been solved completely, consists in finding additional conditions under which a simply-connected Riemann surface will be of hyperbolic or parabolic type (see [[#References|[6]]], [[#References|[7]]], [[#References|[10]]], [[#References|[11]]], and [[Riemann surfaces, classification of|Riemann surfaces, classification of]]).
  
In the case of an arbitrary Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040227.png" />, its [[Universal covering|universal covering]] surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040228.png" /> will always be a simply-connected Riemann surface, and thus belongs to one of the three types above. The Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040229.png" /> itself is considered to be of elliptic, parabolic or hyperbolic type, according to the type of its universal covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040230.png" />. This classification of Riemann surfaces is justified by the following considerations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040231.png" /> be one of three domains: the extended complex plane, the finite complex plane or the open unit disc, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040232.png" /> be some group of Möbius transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040233.png" /> onto itself (automorphisms) without fixed points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040234.png" />. A conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040235.png" /> of the universal covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040236.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040237.png" /> carries the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040238.png" /> of transformations of the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040239.png" />, which is isomorphic to the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040240.png" />, onto some group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040241.png" /> of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040242.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040243.png" /> can be considered as a conformal mapping from the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040244.png" /> onto the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040245.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040246.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040247.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040248.png" /> can be considered as a conformal mapping from the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040249.png" /> onto the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040250.png" /> with some group of automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040251.png" /> isomorphic to the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040252.png" />.
+
In the case of an arbitrary Riemann surface $  R $,  
 +
its [[Universal covering|universal covering]] surface $  \widehat{R}  $
 +
will always be a simply-connected Riemann surface, and thus belongs to one of the three types above. The Riemann surface $  R $
 +
itself is considered to be of elliptic, parabolic or hyperbolic type, according to the type of its universal covering $  \widehat{R}  $.  
 +
This classification of Riemann surfaces is justified by the following considerations. Let $  D $
 +
be one of three domains: the extended complex plane, the finite complex plane or the open unit disc, and let $  \Lambda $
 +
be some group of Möbius transformations of $  D $
 +
onto itself (automorphisms) without fixed points in $  D $.  
 +
A conformal mapping $  w = W( q) $
 +
of the universal covering $  \widehat{R}  $
 +
onto $  D $
 +
carries the group $  \widehat \Lambda  $
 +
of transformations of the covering $  \widehat{R}  $,  
 +
which is isomorphic to the fundamental group $  \pi _ {1} ( R) $,  
 +
onto some group $  \Lambda $
 +
of automorphisms of $  D $.  
 +
Moreover, $  w = W( q) $
 +
can be considered as a conformal mapping from the quotient space $  \widehat{R}  / \widehat \Lambda  $
 +
onto the quotient space $  D/ \Lambda $,  
 +
and $  \widehat{R}  / \widehat \Lambda  $
 +
can be identified with $  R $.  
 +
Thus, $  w = W( q) $
 +
can be considered as a conformal mapping from the Riemann surface $  R $
 +
onto the quotient space $  D/ \Lambda $
 +
with some group of automorphisms $  \Lambda $
 +
isomorphic to the fundamental group $  \pi _ {1} ( R) $.
  
Since a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040253.png" /> of elliptic type is necessarily simply-connected, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040254.png" /> is trivial and thus such a Riemann surface is necessarily the Riemann surface of the function inverse to a rational function. A simply-connected Riemann surface of parabolic type is necessarily the Riemann surface of the function inverse to a meromorphic function in the finite plane. A compact Riemann surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040255.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040256.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040257.png" /> is a Riemann surface of elliptic, parabolic or hyperbolic type, respectively.
+
Since a Riemann surface $  R $
 +
of elliptic type is necessarily simply-connected, the group $  \Lambda $
 +
is trivial and thus such a Riemann surface is necessarily the Riemann surface of the function inverse to a rational function. A simply-connected Riemann surface of parabolic type is necessarily the Riemann surface of the function inverse to a meromorphic function in the finite plane. A compact Riemann surface of genus $  g= 0 $,  
 +
$  g= 1 $
 +
or $  g > 1 $
 +
is a Riemann surface of elliptic, parabolic or hyperbolic type, respectively.
  
In connection with the conformal equivalence of Riemann surfaces there arises the question of the structure of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040258.png" /> of conformal automorphisms of a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040259.png" />. Except for certain simple cases, this group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040260.png" /> is discrete and for compact Riemann surfaces of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040261.png" /> it is finite (Schwarz' theorem). There are only seven exceptional cases in which the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040262.png" /> is continuous, namely (representatives of the corresponding conformal classes are given): the sphere in the elliptic case; the sphere with one or two punctures and the torus in the parabolic case; the open disc, the punctured open disc and an annulus in the hyperbolic case.
+
In connection with the conformal equivalence of Riemann surfaces there arises the question of the structure of the group $  \Sigma $
 +
of conformal automorphisms of a Riemann surface $  R $.  
 +
Except for certain simple cases, this group $  \Sigma $
 +
is discrete and for compact Riemann surfaces of genus $  g > 1 $
 +
it is finite (Schwarz' theorem). There are only seven exceptional cases in which the group $  \Sigma $
 +
is continuous, namely (representatives of the corresponding conformal classes are given): the sphere in the elliptic case; the sphere with one or two punctures and the torus in the parabolic case; the open disc, the punctured open disc and an annulus in the hyperbolic case.
  
Of great importance also is the moduli problem for Riemann surfaces in its different versions (cf. [[Moduli of a Riemann surface|Moduli of a Riemann surface]]; [[Moduli problem|Moduli problem]]). It is the problem of the possible description of the diversity of conformally-inequivalent Riemann surfaces of different types. For example, it is easy to establish the following facts. The set of types of conformally-inequivalent doubly-connected schlichtartig Riemann surfaces (annuli) depends on one real parameter (the modulus) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040263.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040264.png" />; i.e. two annuli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040265.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040266.png" />, are conformally equivalent if and only if the ratios of their radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040267.png" /> coincide. The set of types of conformally-inequivalent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040268.png" />-connected schlichtartig Riemann surfaces for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040269.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040270.png" /> real parameters. The set of types of conformally-inequivalent closed Riemann surfaces of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040271.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040272.png" /> depends on two real parameters and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040273.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040274.png" /> real parameters (see [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]], and also [[#References|[3]]], [[#References|[12]]], [[#References|[13]]], , ; concerning the behaviour of functions of other classes on Riemann surfaces see [[Riemann surfaces, classification of|Riemann surfaces, classification of]]).
+
Of great importance also is the moduli problem for Riemann surfaces in its different versions (cf. [[Moduli of a Riemann surface|Moduli of a Riemann surface]]; [[Moduli problem|Moduli problem]]). It is the problem of the possible description of the diversity of conformally-inequivalent Riemann surfaces of different types. For example, it is easy to establish the following facts. The set of types of conformally-inequivalent doubly-connected schlichtartig Riemann surfaces (annuli) depends on one real parameter (the modulus) $  k $,
 +
$  0 < k < 1 $;  
 +
i.e. two annuli $  0 < r _  \nu  < | z | < R _  \nu  $,  
 +
$  \nu = 1, 2 $,  
 +
are conformally equivalent if and only if the ratios of their radii $  k = ( r _ {1} /R _ {1} ) = ( r _ {2} / R _ {2} ) $
 +
coincide. The set of types of conformally-inequivalent $  n $-
 +
connected schlichtartig Riemann surfaces for $  n > 2 $
 +
depends on $  3n- 6 $
 +
real parameters. The set of types of conformally-inequivalent closed Riemann surfaces of genus $  g \geq  1 $
 +
for $  g= 1 $
 +
depends on two real parameters and for $  g > 1 $
 +
on $  6g- 6 $
 +
real parameters (see [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]], and also [[#References|[3]]], [[#References|[12]]], [[#References|[13]]], , ; concerning the behaviour of functions of other classes on Riemann surfaces see [[Riemann surfaces, classification of|Riemann surfaces, classification of]]).
  
 
An important aspect of the theory of Riemann surfaces is its connection with the concept of uniformization. In general, for a multiple-valued analytic function
 
An important aspect of the theory of Riemann surfaces is its connection with the concept of uniformization. In general, for a multiple-valued analytic function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040275.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= f( z),
 +
$$
  
its Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040276.png" /> provides a geometrical means of uniformization: The multiple-valued relation (2) is replaced by two single-valued relations
+
its Riemann surface $  R _ {f} $
 +
provides a geometrical means of uniformization: The multiple-valued relation (2) is replaced by two single-valued relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040277.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
= F( p),\ \
 +
= g( p),\ \
 +
p \in R _ {f} ,
 +
$$
  
which give a single-valued expression of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040278.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040279.png" /> in the entire domain of definition of the function (2) as a complete analytic function. On the other hand, the approach of K. Weierstrass to the construction of the notion of the complete analytic function of (2) is based on the use of a local uniformizing parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040280.png" /> that allows one to express the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040281.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040282.png" /> analytically as single-valued analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040283.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040284.png" />, locally in a neighbourhood of some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040285.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040286.png" />. The uniformization problem in its simplest classical form is the problem of synthesis of these two ideas. One has to replace the relation (2) in its entire domain of definition by two analytic representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040287.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040288.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040289.png" /> is a uniformizing complex variable with values in some domain of the plane.
+
which give a single-valued expression of $  z $
 +
and $  w $
 +
in the entire domain of definition of the function (2) as a complete analytic function. On the other hand, the approach of K. Weierstrass to the construction of the notion of the complete analytic function of (2) is based on the use of a local uniformizing parameter $  t $
 +
that allows one to express the variables $  z $
 +
and $  w $
 +
analytically as single-valued analytic functions $  z = z( t) $
 +
and $  w = w( t) $,  
 +
locally in a neighbourhood of some point $  ( z _ {0} , w _ {0} ) $,  
 +
$  w _ {0} = f( z _ {0} ) $.  
 +
The uniformization problem in its simplest classical form is the problem of synthesis of these two ideas. One has to replace the relation (2) in its entire domain of definition by two analytic representations $  z = z( t) $,  
 +
$  w = w( t) $,  
 +
where $  t $
 +
is a uniformizing complex variable with values in some domain of the plane.
  
The above-mentioned statement of the possibility of uniformization was established by Koebe and, independently, by Poincaré almost simultaneously in 1907. If the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040290.png" /> of the function (2) is simply connected or schlichtartig, then the uniformization problem is reduced to constructing a conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040291.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040292.png" /> onto the planar domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040293.png" />. The representations (3) then provide the sought uniformization:
+
The above-mentioned statement of the possibility of uniformization was established by Koebe and, independently, by Poincaré almost simultaneously in 1907. If the Riemann surface $  R _ {f} $
 +
of the function (2) is simply connected or schlichtartig, then the uniformization problem is reduced to constructing a conformal mapping $  \phi : R _ {f} \rightarrow D $
 +
from $  R _ {f} $
 +
onto the planar domain $  D $.  
 +
The representations (3) then provide the sought uniformization:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040294.png" /></td> </tr></table>
+
$$
 +
= g[ \phi  ^ {-} 1 ( t)],\ \
 +
= F[ \phi  ^ {-} 1 ( t)],\ \
 +
t \in D.
 +
$$
  
A conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040295.png" /> onto a planar domain exists only for the schlichtartig Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040296.png" /> (the general uniformization theorem).
+
A conformal mapping $  f $
 +
onto a planar domain exists only for the schlichtartig Riemann surfaces $  R _ {f} $(
 +
the general uniformization theorem).
  
In the general case of an arbitrary analytic relation (2), the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040297.png" /> is not schlichtartig but its universal covering surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040298.png" /> is simply connected and, hence, there exists a conformal mapping
+
In the general case of an arbitrary analytic relation (2), the Riemann surface $  R _ {f} $
 +
is not schlichtartig but its universal covering surface $  \widehat{R}  _ {f} $
 +
is simply connected and, hence, there exists a conformal mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040299.png" /></td> </tr></table>
+
$$
 +
\psi : \widehat{R}  _ {f}  \rightarrow  D,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040300.png" /> is one of the already-mentioned domains: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040302.png" /> or the open unit disc. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040303.png" /> is meromorphic on the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040304.png" /> and, hence, it is also meromorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040305.png" />; moreover, it depends only on the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040307.png" />, of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040308.png" />. Thus one obtains a geometrical uniformization in the form
+
where $  D $
 +
is one of the already-mentioned domains: $  \overline{\mathbf C}\; $,  
 +
$  \mathbf C $
 +
or the open unit disc. The function $  w = f( z) $
 +
is meromorphic on the Riemann surface $  \widehat{R}  _ {f} $
 +
and, hence, it is also meromorphic on $  R _ {f} $;  
 +
moreover, it depends only on the projection $  p = p( q) $,  
 +
$  p \in R _ {f} $,  
 +
of a point $  q \in \widehat{R}  _ {f} $.  
 +
Thus one obtains a geometrical uniformization in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040309.png" /></td> </tr></table>
+
$$
 +
= g[ p( q)],\ \
 +
= F[ p( q)],
 +
$$
  
 
and from it the analytic uniformization
 
and from it the analytic uniformization
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040310.png" /></td> </tr></table>
+
$$
 +
= g \{ p[ \psi  ^ {-} 1 ( t) ] \}  = \Psi ( t),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040311.png" /></td> </tr></table>
+
$$
 +
= F \{ p[ \psi  ^ {-} 1 ( t) ] \}  = \Phi ( t),\  t \in D,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040312.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040313.png" /> are expressed as meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040314.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040315.png" /> of a variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040316.png" />. These functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040317.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040318.png" /> are automorphic functions (cf. [[Automorphic function|Automorphic function]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040319.png" /> relative to the group of automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040320.png" /> isomorphic to the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040321.png" /> of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082040/r082040322.png" /> of the uniformizing function (see [[#References|[3]]], [[#References|[7]]], , ).
+
where $  z $
 +
and $  w $
 +
are expressed as meromorphic functions $  \Phi ( t) $
 +
and $  \Psi ( t) $
 +
of a variable $  t \in D $.  
 +
These functions $  \Phi ( t) $
 +
and $  \Psi ( t) $
 +
are automorphic functions (cf. [[Automorphic function|Automorphic function]]) in $  D $
 +
relative to the group of automorphisms $  \Lambda $
 +
isomorphic to the fundamental group $  \pi _ {1} ( R _ {f} ) $
 +
of the Riemann surface $  R _ {f} $
 +
of the uniformizing function (see [[#References|[3]]], [[#References|[7]]], , ).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann, "Collected works" , Dover, reprint (1953) {{MR|}} {{ZBL|0703.01020}} {{ZBL|08.0231.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1–2''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. [S. Stoilov] Stoilow, "Leçons sur les principes topologiques de la théorie des fonctions analytiques" , Gauthier-Villars (1938) {{MR|0082545}} {{ZBL|0121.06103}} {{ZBL|0072.07604}} {{ZBL|0017.37802}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) {{MR|0122987}} {{MR|1530201}} {{MR|0092855}} {{ZBL|0501.30039}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Nevanlinna, "Uniformisierung" , Springer (1967) {{MR|0228671}} {{ZBL|0152.27401}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) {{MR|0065652}} {{ZBL|0059.06901}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> L.I. Volkovyskii, "Investigation of the type problem for a simply-connected Riemann surface" ''Trudy Mat. Inst. Steklov.'' , '''34''' (1950) pp. 3–171 (In Russian) {{MR|0049330}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> L.I. Volkovyskii, "Contempory studies on Riemann surfaces" ''Uspekhi Mat. Nauk'' , '''11''' : 5 (1956) pp. 77–84 (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> S.L. Krushkal', "Quasi-conformal mappings and Riemann surfaces" , Winston &amp; Wiley (1979) (Translated from Russian) {{MR|536488}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> S.L. Krushkal', B.N. Apanasov, N.A. Gusevskii, "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc. (1986) (Translated from Russian) {{MR|0835439}} {{ZBL|0579.30001}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> E.B. Vinberg, O.V. Shvartsman, "Riemann surfaces" ''J. Soviet Math.'' , '''14''' : 1 (1980) pp. 985–1020 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''16''' (1978) pp. 191–245 {{MR|0538254}} {{ZBL|0445.30032}} </TD></TR><TR><TD valign="top">[15a]</TD> <TD valign="top"> L. Bers, "Quasi-conformal mappings and Teichmüller's theorem" R. Nevanlinna (ed.) et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press (1960) pp. 89–119 {{MR|114898}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15b]</TD> <TD valign="top"> L.V. Ahlfors, "The complex analytic structure of the space of closed Riemann surfaces" R. Nevanlinna (ed.) et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press (1960) pp. 45–66 {{MR|0124486}} {{ZBL|0100.28903}} </TD></TR><TR><TD valign="top">[15c]</TD> <TD valign="top"> L. Bers, "Spaces of Riemann surfaces" J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press (1960) pp. 349–361 {{MR|0124484}} {{MR|0111835}} {{ZBL|0116.28803}} {{ZBL|0106.28501}} </TD></TR><TR><TD valign="top">[15d]</TD> <TD valign="top"> L. Bers, "Simultaneous uniformization" ''Bull. Amer. Math. Soc.'' , '''66''' (1960) pp. 94–97 {{MR|0111834}} {{ZBL|0090.05101}} </TD></TR><TR><TD valign="top">[15e]</TD> <TD valign="top"> L. Bers, "Holomorphic differentials as functions of moduli" ''Bull. Amer. Math. Soc.'' , '''67''' (1961) pp. 206–210 {{MR|0122989}} {{ZBL|0102.06702}} </TD></TR><TR><TD valign="top">[15f]</TD> <TD valign="top"> L. Ahlfors, "On quasiconformal mappings" ''J. d'Anal. Math.'' , '''3''' (1954) pp. 1–58; 207–208 {{MR|0064875}} {{ZBL|0057.06506}} </TD></TR><TR><TD valign="top">[16a]</TD> <TD valign="top"> L. Bers, "Uniformization, moduli, and Kleinian groups" ''Bull. London Math. Soc.'' , '''4''' (1972) pp. 257–300 {{MR|0348097}} {{ZBL|0257.32012}} </TD></TR><TR><TD valign="top">[16b]</TD> <TD valign="top"> L. Bers, "The moduli of Kleinian groups" ''Russian Math. Surveys'' , '''29''' : 2 (1974) pp. 88–102 ''Uspekhi Mat. Nauk'' , '''29''' : 2 (1974) pp. 86–102 {{MR|0422691}} {{ZBL|}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> F. Klein, "Riemannschen Flächen" , Springer, reprint (1986)</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> H. Weyl, "The concept of a Riemann surface" , Addison-Wesley (1955) (Translated from German) {{MR|1440406}} {{MR|0166351}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1974) {{MR|0667739}} {{MR|0712145}} {{MR|0279300}} {{MR|0204641}} {{MR|0130368}} {{MR|0124486}} {{MR|0114911}} {{MR|0093583}} {{MR|0083561}} {{MR|1565732}} {{MR|0090649}} {{MR|0055250}} {{MR|0054729}} {{MR|0054055}} {{MR|0043912}} {{MR|0036318}} {{ZBL|0508.01017}} {{ZBL|0445.30001}} {{ZBL|0213.35602}} {{ZBL|0146.30602}} {{ZBL|0196.33801}} {{ZBL|0100.28903}} {{ZBL|0178.08201}} {{ZBL|0114.28101}} {{ZBL|0071.07502}} {{ZBL|0066.32801}} {{ZBL|0052.30503}} {{ZBL|0050.08403}} {{ZBL|0048.05905}} {{ZBL|0048.05904}} {{ZBL|0042.31603}} {{ZBL|0041.41102}} {{ZBL|0029.25802}} {{ZBL|0018.26202}} {{ZBL|0015.36001}} {{ZBL|0006.26204}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> A. Pfluger, "Theorie der Riemannschen Flächen" , Springer (1957) {{MR|0084031}} {{ZBL|0077.07803}} </TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> L. Sario, M. Nakai, "Classification theory of Riemann surfaces" , Springer (1970) {{MR|0264064}} {{ZBL|0199.40603}} </TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top"> M. Heins, "Hardy classes on Riemann surfaces" , Springer (1969) {{MR|0247069}} {{ZBL|0176.03001}} </TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top"> R.C. Gunning, "Lectures on Riemann surfaces" , Princeton Univ. Press (1966) {{MR|0207977}} {{ZBL|0175.36801}} </TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top"> R.C. Gunning, "Lectures on Riemann surfaces: Jacobi varieties" , Princeton Univ. Press (1972) {{MR|0357407}} {{ZBL|0387.32008}} </TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top"> O. Forster, "Lectures on Riemann surfaces" , Springer (1981) (Translated from German) {{MR|0648106}} {{ZBL|0475.30002}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann, "Collected works" , Dover, reprint (1953) {{MR|}} {{ZBL|0703.01020}} {{ZBL|08.0231.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1–2''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. [S. Stoilov] Stoilow, "Leçons sur les principes topologiques de la théorie des fonctions analytiques" , Gauthier-Villars (1938) {{MR|0082545}} {{ZBL|0121.06103}} {{ZBL|0072.07604}} {{ZBL|0017.37802}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) {{MR|0122987}} {{MR|1530201}} {{MR|0092855}} {{ZBL|0501.30039}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Nevanlinna, "Uniformisierung" , Springer (1967) {{MR|0228671}} {{ZBL|0152.27401}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) {{MR|0065652}} {{ZBL|0059.06901}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> L.I. Volkovyskii, "Investigation of the type problem for a simply-connected Riemann surface" ''Trudy Mat. Inst. Steklov.'' , '''34''' (1950) pp. 3–171 (In Russian) {{MR|0049330}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> L.I. Volkovyskii, "Contempory studies on Riemann surfaces" ''Uspekhi Mat. Nauk'' , '''11''' : 5 (1956) pp. 77–84 (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> S.L. Krushkal', "Quasi-conformal mappings and Riemann surfaces" , Winston &amp; Wiley (1979) (Translated from Russian) {{MR|536488}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> S.L. Krushkal', B.N. Apanasov, N.A. Gusevskii, "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc. (1986) (Translated from Russian) {{MR|0835439}} {{ZBL|0579.30001}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> E.B. Vinberg, O.V. Shvartsman, "Riemann surfaces" ''J. Soviet Math.'' , '''14''' : 1 (1980) pp. 985–1020 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''16''' (1978) pp. 191–245 {{MR|0538254}} {{ZBL|0445.30032}} </TD></TR><TR><TD valign="top">[15a]</TD> <TD valign="top"> L. Bers, "Quasi-conformal mappings and Teichmüller's theorem" R. Nevanlinna (ed.) et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press (1960) pp. 89–119 {{MR|114898}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15b]</TD> <TD valign="top"> L.V. Ahlfors, "The complex analytic structure of the space of closed Riemann surfaces" R. Nevanlinna (ed.) et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press (1960) pp. 45–66 {{MR|0124486}} {{ZBL|0100.28903}} </TD></TR><TR><TD valign="top">[15c]</TD> <TD valign="top"> L. Bers, "Spaces of Riemann surfaces" J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press (1960) pp. 349–361 {{MR|0124484}} {{MR|0111835}} {{ZBL|0116.28803}} {{ZBL|0106.28501}} </TD></TR><TR><TD valign="top">[15d]</TD> <TD valign="top"> L. Bers, "Simultaneous uniformization" ''Bull. Amer. Math. Soc.'' , '''66''' (1960) pp. 94–97 {{MR|0111834}} {{ZBL|0090.05101}} </TD></TR><TR><TD valign="top">[15e]</TD> <TD valign="top"> L. Bers, "Holomorphic differentials as functions of moduli" ''Bull. Amer. Math. Soc.'' , '''67''' (1961) pp. 206–210 {{MR|0122989}} {{ZBL|0102.06702}} </TD></TR><TR><TD valign="top">[15f]</TD> <TD valign="top"> L. Ahlfors, "On quasiconformal mappings" ''J. d'Anal. Math.'' , '''3''' (1954) pp. 1–58; 207–208 {{MR|0064875}} {{ZBL|0057.06506}} </TD></TR><TR><TD valign="top">[16a]</TD> <TD valign="top"> L. Bers, "Uniformization, moduli, and Kleinian groups" ''Bull. London Math. Soc.'' , '''4''' (1972) pp. 257–300 {{MR|0348097}} {{ZBL|0257.32012}} </TD></TR><TR><TD valign="top">[16b]</TD> <TD valign="top"> L. Bers, "The moduli of Kleinian groups" ''Russian Math. Surveys'' , '''29''' : 2 (1974) pp. 88–102 ''Uspekhi Mat. Nauk'' , '''29''' : 2 (1974) pp. 86–102 {{MR|0422691}} {{ZBL|}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> F. Klein, "Riemannschen Flächen" , Springer, reprint (1986)</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> H. Weyl, "The concept of a Riemann surface" , Addison-Wesley (1955) (Translated from German) {{MR|1440406}} {{MR|0166351}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1974) {{MR|0667739}} {{MR|0712145}} {{MR|0279300}} {{MR|0204641}} {{MR|0130368}} {{MR|0124486}} {{MR|0114911}} {{MR|0093583}} {{MR|0083561}} {{MR|1565732}} {{MR|0090649}} {{MR|0055250}} {{MR|0054729}} {{MR|0054055}} {{MR|0043912}} {{MR|0036318}} {{ZBL|0508.01017}} {{ZBL|0445.30001}} {{ZBL|0213.35602}} {{ZBL|0146.30602}} {{ZBL|0196.33801}} {{ZBL|0100.28903}} {{ZBL|0178.08201}} {{ZBL|0114.28101}} {{ZBL|0071.07502}} {{ZBL|0066.32801}} {{ZBL|0052.30503}} {{ZBL|0050.08403}} {{ZBL|0048.05905}} {{ZBL|0048.05904}} {{ZBL|0042.31603}} {{ZBL|0041.41102}} {{ZBL|0029.25802}} {{ZBL|0018.26202}} {{ZBL|0015.36001}} {{ZBL|0006.26204}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> A. Pfluger, "Theorie der Riemannschen Flächen" , Springer (1957) {{MR|0084031}} {{ZBL|0077.07803}} </TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> L. Sario, M. Nakai, "Classification theory of Riemann surfaces" , Springer (1970) {{MR|0264064}} {{ZBL|0199.40603}} </TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top"> M. Heins, "Hardy classes on Riemann surfaces" , Springer (1969) {{MR|0247069}} {{ZBL|0176.03001}} </TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top"> R.C. Gunning, "Lectures on Riemann surfaces" , Princeton Univ. Press (1966) {{MR|0207977}} {{ZBL|0175.36801}} </TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top"> R.C. Gunning, "Lectures on Riemann surfaces: Jacobi varieties" , Princeton Univ. Press (1972) {{MR|0357407}} {{ZBL|0387.32008}} </TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top"> O. Forster, "Lectures on Riemann surfaces" , Springer (1981) (Translated from German) {{MR|0648106}} {{ZBL|0475.30002}} </TD></TR></table>
 
 
  
 
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Revision as of 08:11, 6 June 2020


of an analytic function $ w = f( z) $ of a complex variable $ z $

A surface $ R $ such that the complete analytic function $ w = f( z) $, which is, in general, multiple-valued, can be considered as a single-valued analytic function $ w = F( p) $ of a point $ p $ on $ R $.

The concept of a Riemann surface arose in connection with the studies of algebraic functions $ w = f( z) $ defined by an algebraic equation

$$ \tag{1 } a _ {0} ( z) w ^ {m} + a _ {1} ( z) w ^ {m-} 1 + \dots + a _ {m} ( z) = 0, $$

where $ a _ {j} ( z) $, $ j = 0 \dots m $, are polynomials with constant coefficients, $ a _ {0} ( z) \neq 0 $, $ a _ {m} ( z) \neq 0 $. In the works of V. Puiseux (1850–1851) one discovers a clear understanding of multiple-valuedness, characteristic of these functions $ w = f( z) $, when to each value of the variable $ z $, $ m $ values of the variable $ w $ correspond. B. Riemann (1851–1857, see [1]) was the first to show how for any algebraic function one can construct a surface on which this function can be considered as a single-valued rational function of the point. The obtained Riemann surface can be identified with the algebraic curve defined by equation (1). In general, a mutual penetration (sometimes more intensive, sometimes less intensive) of ideas and methods of the theory of functions of a complex variable on the one hand and of algebra and algebraic geometry on the other hand is characteristic of the whole period of further development of the theory of Riemann surfaces, associated with the names of F. Klein, H. Poincaré, P. Koebe, and others. The landmark of this development was the first edition of the book of H. Weyl [18], in which the general concept of an abstract Riemann surface was formulated.

Definition A: A connected topological Hausdorff space $ R $ is called an abstract Riemann surface or, simply, a Riemann surface, if it admits a covering by open sets $ U $ together with a homeomorphism $ \alpha : U \rightarrow D $ corresponding to each set $ U $, where $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $ is the unit disc in the complex $ z $- plane $ \mathbf C $; moreover, if a point $ p \in R $ belongs to $ U $ and $ U ^ \prime $, then the one-to-one correspondence $ z ^ \prime = \alpha ^ \prime \alpha ^ {-} 1 ( z) $ should be a conformal mapping of the first kind in a neighbourhood of the point $ \alpha ( p) \in D $, that is, $ z ^ \prime = \alpha ^ \prime \alpha ^ {-} 1 ( z) $ is a univalent analytic function in a neighbourhood of the point $ \alpha ( p) \in D $. In other words, an abstract Riemann surface is a two-dimensional complex-analytic manifold.

The definition of a Riemann surface with boundary $ \overline{R}\; $ differs from definition $ A $ by the fact that together with the homeomorphisms $ \alpha : U \rightarrow D $, homeomorphisms $ \alpha : U \rightarrow D _ {0} ^ {+} $ are admitted, where $ D _ {0} ^ {+} = \{ {z \in \mathbf C } : {| z | < 1, \mathop{\rm Im} z \geq 0 } \} $ is the unit upper half-disc in $ \mathbf C $; moreover, it is usually assumed that $ \overline{R}\; $ is not already a Riemann surface in the sense of definition $ A $. The points of a Riemann surface with boundary $ \overline{R}\; $ that have neighbourhoods homeomorphic to $ D $ are called interior and the other points, that are mapped to the points of the segment

$$ \{ {z = x + iy \in \mathbf C } : {- 1 < x < 1, y = 0 } \} , $$

form the boundary $ \partial \overline{R}\; $. The set of interior points of $ \overline{R}\; $( the interior of $ \overline{R}\; $) is a Riemann surface in the sense of definition $ A $. Thus, in the case of a Riemann surface with boundary, the boundary is usually considered to be a non-empty set.

A Riemann surface (with boundary) is a triangulable and orientable manifold with a countable base and, hence, it is separable and metrizable. A compact Riemann surface (without boundary) is called a closed Riemann surface; the wider class of finite Riemann surfaces includes the closed Riemann surfaces and the compact Riemann surfaces with a boundary consisting of a finite number of connected components. Non-compact Riemann surfaces with boundary or without it are called open Riemann surfaces. In certain cases it is more convenient to admit in definition $ A $ not only conformal mappings of the first kind but also conformal mappings of the second kind. A Riemann surface with boundary $ \overline{R}\; $( or without it) obtained using such an approach is, generally speaking, not orientable any more, but under the assumption that it be finite it can be conformally imbedded in an orientable closed Riemann surface: the double of $ \overline{R}\; $( see [8], cf. Double of a Riemann surface).

Let an analytic function $ w = f( z) $ be given by one of its regular elements $ ( a, P) = ( a, P( z- a)) $, i.e. by a pair consisting of a point $ a \in \mathbf C $ and a power series

$$ P( z- a) = \sum _ {\nu = 0 } ^ \infty a _ \nu ( z- a) ^ \nu $$

with centre $ a $ and radius of convergence $ r( a) $, $ 0 < r( a) \leq \infty $. Analytic continuation of the element $ ( a, P) $ along all possible paths in the extended plane $ \overline{\mathbf C}\; $ allows one to obtain all regular elements $ ( b, Q) $ of the same type; in their totality they form the complete analytic function, which is also denoted by $ w = f( z) $. Moreover, under analytic continuation elements of a more general nature arise:

$$ ( b, S) = ( b, S(( z- b) ^ {1/n} )), $$

i.e. pairs consisting of a point $ b \in \overline{\mathbf C}\; $ and a generalized power series (a Puiseux series):

$$ S(( z- b) ^ {1/n} ) = \sum _ {\nu = m } ^ \infty b _ {n} ( z- b) ^ {\nu /n } $$

or (in the case when $ b = \infty $ is the point at infinity):

$$ S( z ^ {-} 1/n ) = \sum _ {\nu = m } ^ \infty b _ \nu z ^ {- \nu /n } , $$

where $ m $ is an integer and $ n $ is a positive integer. Moreover, these series converge when $ | z- b | < r( b) $ or $ | z | > r( \infty ) > 0 $, respectively. The generalized elements $ ( b, S) $ or, more precisely, their equivalence classes, form in their totality the analytic image $ A _ {f} $ corresponding to the given analytic function $ w = f( z) $. Among the equivalence classes of elements $ ( b, S) $ that form the analytic image one can distinguish regular ones, when $ n= 1 $, and ramified ones, when $ n> 1 $. The introduction of an appropriate topology on the analytic image $ A _ {f} $ will turn it into the Riemann surface $ R _ {f} $ of the analytic function $ w = f( z) $. This can be achieved, for example, by defining the neighbourhood of an element $ ( b, S) $, $ b \neq \infty $, as the set consisting of the element $ ( b, S) $ itself and all the regular elements $ ( a, P) $ of $ A _ {f} $ for which $ | b- a | < \rho ^ {n} $, $ \rho < r( b) $, and the series $ P( z- a) $ converges to one of the $ n $ determinations of the series $ S(( z- b) ^ {1/n} ) $ in their common domain of definition, i.e.

$$ P( z- a) \equiv S( \epsilon ( z- b) ^ {1/n} ), $$

where $ \epsilon $ is one of the $ n $- th order roots of unity, $ \epsilon ^ {n} = 1 $. The neighbourhood of the element $ ( \infty , S) $ consists of the element $ ( \infty , S) $ itself and all those regular elements $ ( a, P) $ of $ A _ {f} $ for which $ | a | > \rho ^ {-} 1 $, $ \rho < r( b) $, and the series $ P( z- a) $ converges to one of the $ n $ determinations of the series $ S( z ^ {-} 1/n ) $. The space $ R _ {f} $ satisfies all the conditions of definition $ A $.

Thus, to any analytic function $ w = f( z) $ corresponds a Riemann surface $ R _ {f} $ on which this function is a single-valued analytic function $ w = F( p) $ of a point $ p = ( b, S) \in R _ {f} $. This means that in a neighbourhood of any point $ p _ {0} = ( b, S) $ there exists a local uniformizing parameter $ t = ( z- b) ^ {1/n} $ in which $ w $ is represented as a single-valued analytic function $ w = P( t) = F( p) $. In other words, the Riemann surface $ R _ {f} $ of an analytic function is a geometric construct that is used for the global uniformization of a, generally speaking, multiple-valued relation $ w = f( z) $. In a neighbourhood of each point $ p _ {0} = ( b, S) \in R _ {f} $ this relation is uniformized by the two single-valued analytic functions $ z = b+ t ^ {n} $ and $ w = S( t) $. On the other hand, the projection $ \pi : ( b, S) \rightarrow b $ that takes each element $ p _ {0} = ( b, S) \in R _ {f} $ to its centre $ b $ shows that the Riemann surface $ R _ {f} $ of an analytic function is a (ramified) covering surface over the extended complex plane $ \overline{\mathbf C}\; $ or, which is the same, over the Riemann sphere. The projections of the ramified elements $ ( b, S) $ with $ n > 1 $ are branch points of this covering.

At the same time, to each a priori given Riemann surface $ R $ correspond infinitely-many analytic functions $ w = f( z) $ with precisely $ R $ as Riemann surface, $ R _ {f} = R $. For the case of closed Riemann surfaces this statement was formulated and proved already by Riemann in 1851. The central point of the corresponding proof is the construction on $ R $ of harmonic functions with given singularities. The proof given by Riemann was based on an uncritical application of the so-called Dirichlet principle; Koebe (1909) was the first to give a rigorous proof; later there appeared simpler proofs of this fundamental statement, among them those based on the properly applied Dirichlet principle (see, e.g. [3], [4], [17], [18]).

Whatever the orientable topological surface $ S $ may be, one can construct a Riemann surface $ R $ homeomorphic to $ S $, i.e. a Riemann surface of the same topological type as $ S $. Closed Riemann surfaces are topologically completely determined by one number — the genus $ g $, $ 0 \leq g < + \infty $( cf. Genus of a surface). The topological type of such a Riemann surface $ R $ for $ g = 0 $ is a sphere, for $ g = 1 $— a torus, for $ g > 1 $— a generalized torus, or a sphere with $ g $ handles. By cutting a Riemann surface $ R $ of genus $ g = 0 $ along some arc one obtains a digon with the symbol $ s = aa ^ {-} 1 $ as its topological model or normal form, indicating that the points of the sides $ a $ and $ a ^ {-} 1 $ are identified; when $ g \geq 1 $ one has to carry out $ 2g $[[ Canonical sections|canonical sections]] $ a _ {1} , b _ {1} \dots a _ {g} , b _ {g} $, after which one obtains the normal form of a closed Riemann surface $ R $— a polygon with $ 4g $ pairwise-identified sides; the symbol $ s = a _ {1} \dots $ should indicate the order of appearance of the sides. For instance, in Fig. athe normal forms of a sphere for $ g= 0 $ and a torus for $ g= 1 $ are shown together with their symbols.

Figure: r082040a

From the analytic point of view a closed Riemann surface $ R $ is characterized by the fact that it is the Riemann surface of some algebraic function $ w = f( z) $ defined by an algebraic equation (1) of degree $ m $. This Riemann surface $ R $ can be imagined as $ m $ sheets extending over the Riemann sphere and mutually connected in a certain manner at the branch points and along some lines connecting these points (the manner of connecting is determined by the specific form of equation (1)). The genus $ g $ of the Riemann surface $ R $ can in this case be expressed as a function of the number of sheets $ m $ and the orders $ k _ {1} \dots k _ {s} $ of the branch points by the Riemann–Hurwitz formula

$$ g = \sum _ {\nu = 1 } ^ { s } \frac{k _ \nu - 1 }{2} - m + 1. $$

Finite Riemann surfaces $ \overline{R}\; $ are topologically completely characterized by the genus $ g $, $ 0 \leq g < \infty $, and the number $ l $ of connected components of the boundary; their topological type is a sphere with $ g $ handles and $ l $ holes. In the normal form of a finite Riemann surface, the number of sides is not necessarily even, some sides corresponding to components of the boundary that remain free are not identified. The notion of the genus can also be generalized to open Riemann surfaces $ R $, for example by exhausting $ R $ by a sequence $ \{ \overline{R}\; _ \nu \} _ {\nu = 1 } ^ \infty $ of compact Riemann surfaces with boundary $ \overline{R}\; _ \nu $, belonging to $ R $, and such that $ \overline{R}\; _ \nu $ is contained in $ \overline{R}\; _ {\nu + 1 } $, $ \cup _ {\nu = 1 } ^ \infty \overline{R}\; _ \nu = R $. The genus $ g $ of $ R $ is set equal to $ g = \lim\limits _ {\nu \rightarrow \infty } g _ \nu $, where $ g _ \nu $ is the genus of $ \overline{R}\; _ \nu $. This limit exists and is independent of the choice of the exhaustion $ \{ \overline{R}\; _ \nu \} $, $ 0 \leq g \leq + \infty $. However, the genus does not completely define the topological type of an open Riemann surface; the topological types of open Riemann surfaces can be rather diverse. Thus, in Fig.2a, Fig.2b two models with $ g= 0 $ and $ g= 2 $, respectively, are shown.

An important topological characteristic of a Riemann surface $ R $ is the order of connectivity: $ R $ is called simply connected if any simple closed curve in $ R $ can be deformed continuously into a point without getting out of $ R $, i.e. in other words, if the fundamental group of $ R $ is trivial. Otherwise the Riemann surface $ R $ is called multiply connected. The schlichtartig Riemann surfaces form an important class of Riemann surfaces; they are the Riemann surfaces (with boundary or without) that are split by any simple closed curve into non-intersecting parts. For example, in Fig.2a a topological model of a multiply-connected schlichtartig Riemann surface is shown. A schlichtartig Riemann surface necessarily has genus zero. A schlichtartig Riemann surface $ R $ is called $ n $- connected if the minimum number of sections necessary to convert $ R $ to a simply-connected Riemann surface is equal to $ n- 1 $, $ n \geq 1 $( see Fig.2b).

Figure: r082040b

Figure: r082040c

The topological properties of a Riemann surface $ R $ do not completely characterize the analytic properties of $ R $, i.e. the topological properties of $ R $ do not completely characterize the behaviour of functions of different classes on $ R $. In particular, let $ f: R _ {1} \rightarrow R _ {2} $ be a function on the Riemann surface $ R _ {1} $ with values on another Riemann surface $ R _ {2} $. The function $ f $ is called analytic on $ R _ {1} $ if for any point $ p _ {0} \in R _ {1} $, $ f( p _ {0} ) = q _ {0} $, one can find local uniformizing parameters $ t = \phi ( p) $ in a neighbourhood of $ p _ {0} $ on $ R _ {1} $ and $ \tau = \psi ( q) $ in a neighbourhood of $ q _ {0} $ on $ R _ {2} $, respectively, such that the composite function

$$ \tau = \psi \{ f[ \phi ^ {-} 1 ( t)] \} = g( t) $$

is an analytic function of the complex variable $ t $ in a neighbourhood of the value $ t _ {0} = \phi ( p _ {0} ) $. Two Riemann surfaces $ R _ {1} $ and $ R _ {2} $ are called conformally equivalent, or are said to belong to the same conformal class (cf. Riemann surfaces, conformal classes of), if there exists an analytic function $ f: R _ {1} \rightarrow R _ {2} $ that gives a one-to-one mapping from $ R _ {1} $ onto $ R _ {2} $. From the point of view of the behaviour of analytic functions on Riemann surfaces, conformally-equivalent Riemann surfaces are considered as one and the same Riemann surface, but topologically-equivalent Riemann surfaces are not always conformally equivalent.

In terms of Riemann surfaces, the Riemann mapping theorem can be formulated as follows: Any simply-connected Riemann surface is conformally equivalent to one of the following three domains: 1) the extended complex plane $ \overline{\mathbf C}\; = \mathbf C \cup \{ \infty \} $, i.e. the Riemann sphere (the elliptic case); 2) the finite complex plane $ \mathbf C $, i.e. the punctured Riemann sphere (the parabolic case); or 3) the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $ in $ \mathbf C $, i.e. the Riemann sphere with a section of positive length (the hyperbolic case). An important result is that any schlichtartig Riemann surface is conformally equivalent to some canonical domain in the extended complex plane. As such a canonical domain one may take the entire extended plane with a finite or infinite number of sections parallel to the real axis; moreover, some of these sections may degenerate into points. As mentioned above, in the case of a simply-connected Riemann surface the canonical domain either has no sections (elliptic type), or the section degenerates into a point (parabolic type), or the section has a positive length (hyperbolic type). All three types of simply-connected Riemann surfaces are conformally different, although the last two of them are topologically equivalent. The problem of types, which has not yet (1991) been solved completely, consists in finding additional conditions under which a simply-connected Riemann surface will be of hyperbolic or parabolic type (see [6], [7], [10], [11], and Riemann surfaces, classification of).

In the case of an arbitrary Riemann surface $ R $, its universal covering surface $ \widehat{R} $ will always be a simply-connected Riemann surface, and thus belongs to one of the three types above. The Riemann surface $ R $ itself is considered to be of elliptic, parabolic or hyperbolic type, according to the type of its universal covering $ \widehat{R} $. This classification of Riemann surfaces is justified by the following considerations. Let $ D $ be one of three domains: the extended complex plane, the finite complex plane or the open unit disc, and let $ \Lambda $ be some group of Möbius transformations of $ D $ onto itself (automorphisms) without fixed points in $ D $. A conformal mapping $ w = W( q) $ of the universal covering $ \widehat{R} $ onto $ D $ carries the group $ \widehat \Lambda $ of transformations of the covering $ \widehat{R} $, which is isomorphic to the fundamental group $ \pi _ {1} ( R) $, onto some group $ \Lambda $ of automorphisms of $ D $. Moreover, $ w = W( q) $ can be considered as a conformal mapping from the quotient space $ \widehat{R} / \widehat \Lambda $ onto the quotient space $ D/ \Lambda $, and $ \widehat{R} / \widehat \Lambda $ can be identified with $ R $. Thus, $ w = W( q) $ can be considered as a conformal mapping from the Riemann surface $ R $ onto the quotient space $ D/ \Lambda $ with some group of automorphisms $ \Lambda $ isomorphic to the fundamental group $ \pi _ {1} ( R) $.

Since a Riemann surface $ R $ of elliptic type is necessarily simply-connected, the group $ \Lambda $ is trivial and thus such a Riemann surface is necessarily the Riemann surface of the function inverse to a rational function. A simply-connected Riemann surface of parabolic type is necessarily the Riemann surface of the function inverse to a meromorphic function in the finite plane. A compact Riemann surface of genus $ g= 0 $, $ g= 1 $ or $ g > 1 $ is a Riemann surface of elliptic, parabolic or hyperbolic type, respectively.

In connection with the conformal equivalence of Riemann surfaces there arises the question of the structure of the group $ \Sigma $ of conformal automorphisms of a Riemann surface $ R $. Except for certain simple cases, this group $ \Sigma $ is discrete and for compact Riemann surfaces of genus $ g > 1 $ it is finite (Schwarz' theorem). There are only seven exceptional cases in which the group $ \Sigma $ is continuous, namely (representatives of the corresponding conformal classes are given): the sphere in the elliptic case; the sphere with one or two punctures and the torus in the parabolic case; the open disc, the punctured open disc and an annulus in the hyperbolic case.

Of great importance also is the moduli problem for Riemann surfaces in its different versions (cf. Moduli of a Riemann surface; Moduli problem). It is the problem of the possible description of the diversity of conformally-inequivalent Riemann surfaces of different types. For example, it is easy to establish the following facts. The set of types of conformally-inequivalent doubly-connected schlichtartig Riemann surfaces (annuli) depends on one real parameter (the modulus) $ k $, $ 0 < k < 1 $; i.e. two annuli $ 0 < r _ \nu < | z | < R _ \nu $, $ \nu = 1, 2 $, are conformally equivalent if and only if the ratios of their radii $ k = ( r _ {1} /R _ {1} ) = ( r _ {2} / R _ {2} ) $ coincide. The set of types of conformally-inequivalent $ n $- connected schlichtartig Riemann surfaces for $ n > 2 $ depends on $ 3n- 6 $ real parameters. The set of types of conformally-inequivalent closed Riemann surfaces of genus $ g \geq 1 $ for $ g= 1 $ depends on two real parameters and for $ g > 1 $ on $ 6g- 6 $ real parameters (see Riemann surfaces, conformal classes of, and also [3], [12], [13], , ; concerning the behaviour of functions of other classes on Riemann surfaces see Riemann surfaces, classification of).

An important aspect of the theory of Riemann surfaces is its connection with the concept of uniformization. In general, for a multiple-valued analytic function

$$ \tag{2 } w = f( z), $$

its Riemann surface $ R _ {f} $ provides a geometrical means of uniformization: The multiple-valued relation (2) is replaced by two single-valued relations

$$ \tag{3 } w = F( p),\ \ z = g( p),\ \ p \in R _ {f} , $$

which give a single-valued expression of $ z $ and $ w $ in the entire domain of definition of the function (2) as a complete analytic function. On the other hand, the approach of K. Weierstrass to the construction of the notion of the complete analytic function of (2) is based on the use of a local uniformizing parameter $ t $ that allows one to express the variables $ z $ and $ w $ analytically as single-valued analytic functions $ z = z( t) $ and $ w = w( t) $, locally in a neighbourhood of some point $ ( z _ {0} , w _ {0} ) $, $ w _ {0} = f( z _ {0} ) $. The uniformization problem in its simplest classical form is the problem of synthesis of these two ideas. One has to replace the relation (2) in its entire domain of definition by two analytic representations $ z = z( t) $, $ w = w( t) $, where $ t $ is a uniformizing complex variable with values in some domain of the plane.

The above-mentioned statement of the possibility of uniformization was established by Koebe and, independently, by Poincaré almost simultaneously in 1907. If the Riemann surface $ R _ {f} $ of the function (2) is simply connected or schlichtartig, then the uniformization problem is reduced to constructing a conformal mapping $ \phi : R _ {f} \rightarrow D $ from $ R _ {f} $ onto the planar domain $ D $. The representations (3) then provide the sought uniformization:

$$ z = g[ \phi ^ {-} 1 ( t)],\ \ w = F[ \phi ^ {-} 1 ( t)],\ \ t \in D. $$

A conformal mapping $ f $ onto a planar domain exists only for the schlichtartig Riemann surfaces $ R _ {f} $( the general uniformization theorem).

In the general case of an arbitrary analytic relation (2), the Riemann surface $ R _ {f} $ is not schlichtartig but its universal covering surface $ \widehat{R} _ {f} $ is simply connected and, hence, there exists a conformal mapping

$$ \psi : \widehat{R} _ {f} \rightarrow D, $$

where $ D $ is one of the already-mentioned domains: $ \overline{\mathbf C}\; $, $ \mathbf C $ or the open unit disc. The function $ w = f( z) $ is meromorphic on the Riemann surface $ \widehat{R} _ {f} $ and, hence, it is also meromorphic on $ R _ {f} $; moreover, it depends only on the projection $ p = p( q) $, $ p \in R _ {f} $, of a point $ q \in \widehat{R} _ {f} $. Thus one obtains a geometrical uniformization in the form

$$ z = g[ p( q)],\ \ w = F[ p( q)], $$

and from it the analytic uniformization

$$ z = g \{ p[ \psi ^ {-} 1 ( t) ] \} = \Psi ( t), $$

$$ w = F \{ p[ \psi ^ {-} 1 ( t) ] \} = \Phi ( t),\ t \in D, $$

where $ z $ and $ w $ are expressed as meromorphic functions $ \Phi ( t) $ and $ \Psi ( t) $ of a variable $ t \in D $. These functions $ \Phi ( t) $ and $ \Psi ( t) $ are automorphic functions (cf. Automorphic function) in $ D $ relative to the group of automorphisms $ \Lambda $ isomorphic to the fundamental group $ \pi _ {1} ( R _ {f} ) $ of the Riemann surface $ R _ {f} $ of the uniformizing function (see [3], [7], , ).

References

[1] B. Riemann, "Collected works" , Dover, reprint (1953) Zbl 0703.01020 Zbl 08.0231.03
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Comments

For the generalization to "Riemann surfaces over Cn" see, e.g., [a6] and Riemannian domain.

References

[a1] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a2] P. Griffiths, "Introduction to algebraic curves" , Amer. Math. Soc. (1989) MR1013999 Zbl 0696.14012
[a3] H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6 MR0583745 Zbl 0475.30001
[a4] H. Behnke, F. Sommer, "Theorie der analytischen Funktionen einer komplexen Veränderlichen" , Springer (1955) MR0073682 Zbl 0065.06102
[a5] H. Cohn, "Conformal mapping on Riemann surfaces" , Dover, reprint (1980) MR0594937 Zbl 0493.30001
[a6] H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) pp. Chapt. VI (Elraged & Revised Edition. Original: 1934)
[a7] W. Osgood, "Lehrbuch der Funktionentheorie" , 1–2 , Chelsea, reprint (1965) MR0196039 MR0196038 Zbl 0005.29904 Zbl 58.0390.04 Zbl 54.0326.10 Zbl 55.0171.02 Zbl 50.0209.04 Zbl 43.0476.02 Zbl 37.0409.02 Zbl 38.0412.01
How to Cite This Entry:
Riemann surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_surface&oldid=23963
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article