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)