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


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 $.


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|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]]).

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|Stein space]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Hörmander,   "An introduction to complex analysis in several variables" , North-Holland  (1973)</TD></TR></table>



====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|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 $[[#References|[a1]]].

====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>