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''Riemann domain, complex (-analytic) manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r0821405.png" />''
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An analogue of the [[Riemann surface|Riemann surface]] of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r0821406.png" /> of a single complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r0821407.png" /> for the case of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r0821408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r0821409.png" />, of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214011.png" />.
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''Riemann domain, complex (-analytic) manifold over  $  \mathbf C ^{n} $''
  
More precisely, a path-connected Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214012.png" /> is called an (abstract) Riemann domain if there is a local homeomorphism (a projection) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214013.png" /> such that for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214014.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214015.png" /> that transforms homeomorphically into a polydisc
 
  
<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/r082140/r08214016.png" /></td> </tr></table>
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An analogue of the [[Riemann surface|Riemann surface]] of an analytic function  $  w = f(z) $
 +
of a single complex variable  $  z $
 +
for the case of analytic functions  $  w = f(z) $,
 +
$  z = (z _{1} \dots z _{n} ) $,
 +
of several complex variables  $  z _{1} \dots z _{n} $,
 +
$  n \geq 2 $.
  
<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/r082140/r08214017.png" /></td> </tr></table>
 
  
in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214018.png" />. A Riemann domain is a separable space.
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More precisely, a path-connected Hausdorff space $  R $
 +
is called an (abstract) Riemann domain if there is a local homeomorphism (a projection)  $  \pi : \  R \rightarrow \mathbf C ^{n} $
 +
such that for each point  $  p _{0} \in R $
 +
there is a neighbourhood  $  U(p _{0} ; \  \epsilon ) $
 +
that transforms homeomorphically into a polydisc
  
A complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214019.png" /> is called holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214020.png" /> if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214021.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214023.png" /> complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214024.png" /> is holomorphic in the corresponding polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214025.png" />. The projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214026.png" /> is given by the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214027.png" /> holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214028.png" />, which correspond to coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214030.png" />. Starting from a given regular element of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214031.png" />, its Riemann domain is constructed in the same way as the Riemann surface of a given analytic function of one complex variable, i.e. initially by means of analytic continuation one constructs the [[Complete analytic function|complete analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214032.png" />, and then, using neighbourhoods, one introduces a topology into the set of elements of the complete analytic function. Like Riemann surfaces, Riemann domains arise unavoidably in connection with analytic continuation of a given element of an analytic function when, following the ideas of B. Riemann, one tries to represent the complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214033.png" /> as a single-valued point function on a domain.
+
$$
 +
D(z ^{0} ; \  \epsilon )\  =
 +
$$
 +
 
 +
 
 +
$$
 +
= \
 +
\{ {z = (z _{1} \dots z _{n} ) \in \mathbf C ^ n} : {
 +
| z _{j} - z _{j} ^{0} | < \epsilon ,\  j = 1 \dots n} \}
 +
$$
 +
 
 +
 
 +
in the complex space  $  \mathbf C ^{n} $.  
 +
A Riemann domain is a separable space.
 +
 
 +
A complex function  $  g $
 +
is called holomorphic on $  R $
 +
if for any point $  p _{0} \in R $
 +
the function $  g[ \pi ^{-1} (z)] $
 +
of $  n $
 +
complex variables $  z _{1} \dots z _{n} $
 +
is holomorphic in the corresponding polydisc $  D(z ^{0} ; \  \epsilon ) $.  
 +
The projection $  \pi $
 +
is given by the choice of $  n $
 +
holomorphic functions $  \pi = ( \pi _{1} \dots \pi _{n} ) $,  
 +
which correspond to coordinates $  z _{1} \dots z _{n} $
 +
in $  \mathbf C ^{n} $.  
 +
Starting from a given regular element of an analytic function $  w = f(z) $,  
 +
its Riemann domain is constructed in the same way as the Riemann surface of a given analytic function of one complex variable, i.e. initially by means of analytic continuation one constructs the [[Complete analytic function|complete analytic function]] $  w = f(z) $,  
 +
and then, using neighbourhoods, one introduces a topology into the set of elements of the complete analytic function. Like Riemann surfaces, Riemann domains arise unavoidably in connection with analytic continuation of a given element of an analytic function when, following the ideas of B. Riemann, one tries to represent the complete analytic function $  w = f(z) $
 +
as a single-valued point function on a domain.
  
 
In particular, Riemann domains arise as multi-sheeted domains of holomorphy of analytic functions of several complex variables. Oka's theorem states that a Riemann domain is a [[Domain of holomorphy|domain of holomorphy]] if and only if it is holomorphically convex (see [[Holomorphically-convex complex space|Holomorphically-convex complex space]]).
 
In particular, Riemann domains arise as multi-sheeted domains of holomorphy of analytic functions of several complex variables. Oka's theorem states that a Riemann domain is a [[Domain of holomorphy|domain of holomorphy]] if and only if it is holomorphically convex (see [[Holomorphically-convex complex space|Holomorphically-convex complex space]]).
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====Comments====
 
====Comments====
The notion as presented above of a Riemann domain has been extended in several ways: Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214034.png" /> one may choose any (model) complex-analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214035.png" /> (cf. [[Complex space|Complex space]]). An unramified Riemann domain over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214036.png" /> is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214037.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214038.png" /> is a complex-analytic space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214039.png" /> is a locally biholomorphic mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214040.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214041.png" />.
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The notion as presented above of a Riemann domain has been extended in several ways: Instead of $  \mathbf C ^{n} $
 +
one may choose any (model) complex-analytic space $  S $(
 +
cf. [[Complex space|Complex space]]). An unramified Riemann domain over $  S $
 +
is a triple $  ( R,\  \Phi ,\  S ) $
 +
where $  R $
 +
is a complex-analytic space and $  \Phi $
 +
is a locally biholomorphic mapping from $  R $
 +
into $  S $.
 +
 
  
Next, a ramified Riemann domain over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214042.png" /> is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214043.png" /> where again <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214044.png" /> is a complex-analytic space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214045.png" /> is now a discrete open holomorphic mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214046.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082140/r08214047.png" /> [[#References|[a1]]].
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Next, a ramified Riemann domain over $  S $
 +
is a triple $  (R ,\  \Phi ,\  S ) $
 +
where again $  R $
 +
is a complex-analytic space and $  \Phi $
 +
is now a discrete open holomorphic mapping from $  R $
 +
to $  S $[[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Behnke,  P. Thullen,  "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer  (1970)  pp. Chapt. VI  (Elraged &amp; Revised Edition. Original: 1934)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  K. Fritzsche,  "Several complex variables" , Springer  (1976)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Behnke,  P. Thullen,  "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer  (1970)  pp. Chapt. VI  (Elraged &amp; Revised Edition. Original: 1934)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  K. Fritzsche,  "Several complex variables" , Springer  (1976)  (Translated from German)</TD></TR></table>

Latest revision as of 19:58, 28 January 2020


Riemann domain, complex (-analytic) manifold over $ \mathbf C ^{n} $


An analogue of the Riemann surface of an analytic function $ w = f(z) $ of a single complex variable $ z $ for the case of analytic functions $ w = f(z) $, $ z = (z _{1} \dots z _{n} ) $, of several complex variables $ z _{1} \dots z _{n} $, $ n \geq 2 $.


More precisely, a path-connected Hausdorff space $ R $ is called an (abstract) Riemann domain if there is a local homeomorphism (a projection) $ \pi : \ R \rightarrow \mathbf C ^{n} $ such that for each point $ p _{0} \in R $ there is a neighbourhood $ U(p _{0} ; \ \epsilon ) $ that transforms homeomorphically into a polydisc

$$ D(z ^{0} ; \ \epsilon )\ = $$


$$ = \ \{ {z = (z _{1} \dots z _{n} ) \in \mathbf C ^ n} : { | z _{j} - z _{j} ^{0} | < \epsilon ,\ j = 1 \dots n} \} $$


in the complex space $ \mathbf C ^{n} $. A Riemann domain is a separable space.

A complex function $ g $ is called holomorphic on $ R $ if for any point $ p _{0} \in R $ the function $ g[ \pi ^{-1} (z)] $ of $ n $ complex variables $ z _{1} \dots z _{n} $ is holomorphic in the corresponding polydisc $ D(z ^{0} ; \ \epsilon ) $. The projection $ \pi $ is given by the choice of $ n $ holomorphic functions $ \pi = ( \pi _{1} \dots \pi _{n} ) $, which correspond to coordinates $ z _{1} \dots z _{n} $ in $ \mathbf C ^{n} $. Starting from a given regular element of an analytic function $ w = f(z) $, its Riemann domain is constructed in the same way as the Riemann surface of a given analytic function of one complex variable, i.e. initially by means of analytic continuation one constructs the complete analytic function $ w = f(z) $, and then, using neighbourhoods, one introduces a topology into the set of elements of the complete analytic function. Like Riemann surfaces, Riemann domains arise unavoidably in connection with analytic continuation of a given element of an analytic function when, following the ideas of B. Riemann, one tries to represent the complete analytic function $ w = f(z) $ as a single-valued point function on a domain.

In particular, Riemann domains arise as multi-sheeted domains of holomorphy of analytic functions of several complex variables. Oka's theorem states that a Riemann domain is a domain of holomorphy if and only if it is holomorphically convex (see Holomorphically-convex complex space).

Modern studies of Riemann domains are conducted within the framework of the general theory of analytic spaces. A generalization of the concept of a domain of holomorphy leads to Stein spaces (cf. Stein space).

References

[1] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[2] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)
[3] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)


Comments

The notion as presented above of a Riemann domain has been extended in several ways: Instead of $ \mathbf C ^{n} $ one may choose any (model) complex-analytic space $ S $( cf. Complex space). An unramified Riemann domain over $ S $ is a triple $ ( R,\ \Phi ,\ S ) $ where $ R $ is a complex-analytic space and $ \Phi $ is a locally biholomorphic mapping from $ R $ into $ S $.


Next, a ramified Riemann domain over $ S $ is a triple $ (R ,\ \Phi ,\ S ) $ where again $ R $ is a complex-analytic space and $ \Phi $ is now a discrete open holomorphic mapping from $ R $ to $ S $[a1].

References

[a1] H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) pp. Chapt. VI (Elraged & Revised Edition. Original: 1934)
[a2] H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) (Translated from German)
How to Cite This Entry:
Riemannian domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_domain&oldid=12703
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article