generic manifold

Definition.

Let $M\subset {ℂ}^N$ be a real submanifold of real dimension $n$. We say that $M$ is a generic manifold if for every $x\in M$ we have

$$T_x(M)+J{T}_x(M)=T_x({ℂ}^N),$$

where $J$ denotes the operator of multiplication by the imaginary unit in$T_x({ℂ}^N)$. That is every vector in$T_x({ℂ}^N)$ can be written as $X+JY$ where $X,Y\in T_x(M)$.

For more details about the tangent spaces and the $J$ operator see theentry onCR manifolds (http://planetmath.org/CRSubmanifold). In fact every generic manifold isalso CR manifold (the converse is not true however). A basic important resultabout generic submanifolds is.

Theorem.

Let $M\mathrm{\subset }{\mathrm{C}}^N$ be a generic submanifold and let$f\mathrm{:}U\mathrm{\subset }{\mathrm{C}}^N\mathrm{\to }\mathrm{C}$ be a holomorphic functionwhere $U$ is a connected open set such that $M\mathrm{\cap }U\mathrm{\ne }\mathrm{\varnothing }$, and furthersuppose that $f\mathit{}\mathrm{(}M\mathrm{\cap }U\mathrm{)}\mathrm{=}\mathrm{\{}\mathrm{0}\mathrm{\}}$, that is $f$ is zero when restrictedto $M$. Then in fact $f\mathrm{\equiv }\mathrm{0}$ on $U$.

For example in ${ℂ}^1$ the real line is a generic submanifold, and any holomorphic function which is zero on the real line is zero everywhere (if thedomain 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.