generic manifold


Let MN be a real submanifold of real dimension n. We say that M is a generic manifold if for every xM we have


where J denotes the operator of multiplication by the imaginary unitMathworldPlanetmath in Tx(N). That is every vector in Tx(N) can be written as X+JY where X,YTx(M).

For more details about the tangent spaces and the J operator see the entry on CR manifolds ( In fact every generic manifold is also CR manifold (the converse is not true however). A basic important result about generic submanifolds is.


Let MCN be a generic submanifold and let f:UCNC be a holomorphic functionMathworldPlanetmath where U is a connected open set such that MU, and further suppose that f(MU)={0}, that is f is zero when restricted to M. Then in fact f0 on U.

For example in 1 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 planeMathworldPlanetmath so the above is mostly useful for higher dimensions.


  • 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