Covering domain

domain over $\mathbf{C}^n$

A pair $(X,\pi)$, where $X$ is an arcwise-connected Hausdorff space and $\pi$ is a local homeomorphism, called a projection. Covering domains are encountered in the analytic continuation of holomorphic functions. For every analytic (possibly multivalent) function $f$ in a domain $D \subset \mathbf{C}^n$ there is a corresponding covering domain $\tilde D$ with a projection $\pi : \tilde D \rightarrow D$, just as for every analytic function of one complex variable there is a corresponding Riemann surface; the function $f$ is single-valued on $\tilde D$. Covering domains are also called Riemann domains.

#### References[edit]

[1] |  B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)

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#### Comments[edit]

A covering domain is sometimes called a manifold spread over $\mathbf{C}^n$. See also Domain of holomorphy; Riemannian domain; Holomorphic envelope.

#### References[edit]

[a1] |  R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Section G

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[a2] |  H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German)