Stein manifold
Definition.
A complex manifold $M$ of complex dimension $n$ is a Stein manifold^{} if it satisfies the following properties

1.
$M$ is holomorphically convex,

2.
if $z,w\in M$ and $z\ne w$ then $f(z)\ne f(w)$ for some function $f$ holomorphic on $M$ (i.e. $M$ is holomorphically separable),

3.
for every $z\in M$ there are holomorphic functions ${f}_{1},\mathrm{\dots},{f}_{n}$ which form a coordinate system^{} at $z$ (i.e. $M$ is holomorphically spreadable).
Stein manifold is a generalization of the concept of the domain of holomorphy to manifolds. Furthermore, Stein manifolds are the generalizations of Riemann surfaces in higher dimensions. Every noncompact Riemann surface is a Stein manifold by a theorem of Behnke and Stein. Note that every domain of holomorphy in ${\u2102}^{n}$ is a Stein manifold. It is not hard to see that every closed complex submanifold of a Stein manifold is Stein.
Theorem (Remmert, Narasimhan, Bishop).
If $M$ is a Stein manifold of dimension $n$. There exists a proper (http://planetmath.org/ProperMap) holomorphic embedding of $M$ into ${\mathrm{C}}^{\mathrm{2}\mathit{}n\mathrm{+}\mathrm{1}}$.
Note that no compact complex manifold can be Stein since compact complex manifolds have no holomorphic functions. On the other hand, every compact complex manifold is holomorphically convex.
References
 1 Lars Hörmander. , NorthHolland Publishing Company, New York, New York, 1973.
 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title  Stein manifold 

Canonical name  SteinManifold 
Date of creation  20130322 15:04:37 
Last modified on  20130322 15:04:37 
Owner  jirka (4157) 
Last modified by  jirka (4157) 
Numerical id  7 
Author  jirka (4157) 
Entry type  Definition 
Classification  msc 32E10 
Related topic  HolomorphicallyConvex 
Related topic  DomainOfHolomorphy 
Defines  holomorphically separable 
Defines  holomorphically spreadable 