A paracompact complex analytic manifold with the following properties:
1) for any compact set the set
where is the algebra of holomorphic functions on , is compact (holomorphic convexity);
2) for any two different points there is a function such that (holomorphic separability);
3) in a neighbourhood of any point there is a holomorphic chart whose coordinate functions belong to .
The requirement of holomorphic convexity can be replaced by the following one: For any sequence without limit points there is a function such that .
The class of Stein manifolds was introduced by K. Stein  as a natural generalization of the notion of a domain of holomorphy in . Any closed analytic submanifold in is a Stein manifold; conversely, any -dimensional Stein manifold has a proper holomorphic imbedding in (cf. Proper morphism). Any non-compact Riemann surface is a Stein manifold. The direct generalization of a Stein manifold is a Stein space.
See also the references to Stein space.
|||K. Stein, "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem" Math. Ann. , 123 (1951) pp. 201–222|
Stein manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stein_manifold&oldid=18400