They were introduced by and named after Karl Stein (1951).
denote the ring of holomorphic functions on
a Stein manifold if the following conditions hold: Let X be a connected, non-compact Riemann surface.
Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial.
Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" holomorphic functions taking values in the complex numbers.
See for example Cartan's theorems A and B, relating to sheaf cohomology.
The initial impetus was to have a description of the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.
In the GAGA set of analogies, Stein manifolds correspond to affine varieties.
Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves.
It is known that a Stein manifold is elliptic if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory".
[2][3] Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.