Riemannian domain

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)