generic manifold
Definition.
Let be a real submanifold of real dimension . We say that is a generic manifold if for every we have
where denotes the operator of multiplication by the imaginary unit in
. That is every vector in
can be written as where .
For more details about the tangent spaces and the operator see the entry on CR manifolds (http://planetmath.org/CRSubmanifold). In fact every generic manifold is also CR manifold (the converse is not true however). A basic important result about generic submanifolds is.
Theorem.
Let be a generic submanifold and let
be a holomorphic function
where is a connected open set such that , and further
suppose that , that is is zero when restricted
to . Then in fact on .
For example in the real line is a generic submanifold, and any holomorphic function which is zero on the real line is zero everywhere (if the
domain of the function is connected and intersects the real line of course). There are of course much stronger uniqueness results for the complex plane so the above is mostly useful for higher dimensions.
References
- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
Title | generic manifold |
---|---|
Canonical name | GenericManifold |
Date of creation | 2013-03-22 14:56:03 |
Last modified on | 2013-03-22 14:56:03 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 5 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 32V05 |
Synonym | generic submanifold |
Related topic | CRSubmanifold |
Related topic | TotallyRealSubmanifold |
Related topic | TangentialCauchyRiemannComplexOfCinftySmoothForms |
Related topic | ACRcomplex |