It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.
The embedding theorem for Stein manifolds states the following: Every Stein manifold of complex dimension can be embedded into by a biholomorphicproper map.
These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the ambient space (because the embedding is biholomorphic).
Every Stein manifold of (complex) dimension n has the homotopy type of an n-dimensional CW-Complex.
In one complex dimension the Stein condition can be simplified: a connected Riemann surface is a Stein manifold if and only if it is not compact. This can be proved using a version of the Runge theorem for Riemann surfaces, due to Behnke and Stein.
Every Stein manifold is holomorphically spreadable, i.e. for every point , there are holomorphic functions defined on all of which form a local coordinate system when restricted to some open neighborhood of .
Being a Stein manifold is equivalent to being a (complex) strongly pseudoconvex manifold. The latter means that it has a strongly pseudoconvex (or plurisubharmonic) exhaustive function, i.e. a smooth real function on (which can be assumed to be a Morse function) with , such that the subsets are compact in for every real number . This is a solution to the so-called Levi problem, named after E. E. Levi (1911). The function invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domains. A Stein domain is the preimage . Some authors call such manifolds therefore strictly pseudoconvex manifolds.
Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface X with a real-valued Morse function f on X such that, away from the critical points of f, the field of complex tangencies to the preimage is a contact structure that induces an orientation on Xc agreeing with the usual orientation as the boundary of That is, is a Stein filling of Xc.
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".
Relation to smooth manifolds
Every compact smooth manifold of dimension 2n, which has only handles of index Thurston-Bennequin framing). Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.
Forster, Otto (1981), Lectures on Riemann surfaces, Graduate Text in Mathematics, 81, New-York: Springer Verlag, ISBN0-387-90617-7 (including a proof of Behnke-Stein and Grauert-Röhrl theorems)
Hörmander, Lars (1990), An introduction to complex analysis in several variables, North-Holland Mathematical Library, 7, Amsterdam: North-Holland Publishing Co., ISBN978-0-444-88446-6, MR1045639 (including a proof of the embedding theorem)
Grauert, Hans; Remmert, Reinhold (1979), Theory of Stein spaces, Grundlehren der Mathematischen Wissenschaften, 236, Berlin-New York: Springer-Verlag, ISBN3-540-90388-7, MR0580152
Stein, Karl (1951), "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem", Math. Ann. (in German), 123: 201-222, doi:10.1007/bf02054949, MR0043219
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