hypersurfaceHypersurface2004-08-11 13:42:302007-12-18 12:16:00Definitionsmooth hypersurfacereal analytic hypersurfacereal hypersurfacelocal defining functionsingular hypersurfacenon-singular hypersurfacehypervariety% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}\begin{defn}
Let $M$ be a subset of ${\mathbb{R}}^n$ such that for every point
$p \in M$ there exists a neighbourhood $U_p$ of $p$ in ${\mathbb{R}}^n$
and a continuously differentiable function $\rho \colon U \to {\mathbb{R}}$ with
$\operatorname{grad} \rho \not= 0$ on $U$,
such that
\begin{equation*}
M \cap U = \{ x \in U \mid \rho(x) = 0 \} .
\end{equation*}
Then $M$ is called a {\em hypersurface}.
\end{defn}
If $\rho$ is in fact smooth then $M$ is a {\em smooth hypersurface} and
similarly if $\rho$ is real analytic then $M$ is a {\em real analytic
hypersurface}. If
we identify ${\mathbb{R}}^{2n}$ with ${\mathbb{C}}^n$ and we have a
hypersurface there it is called a {\em real hypersurface} in
${\mathbb{C}}^n$. $\rho$ is usually called the {\em local defining function}.
Hypersurface is really special name for a submanifold of codimension 1. In fact if $M$ is just a topological manifold of codimension 1, then it is often also called a hypersurface.
A \PMlinkname{real}{RealAnalyticSubvariety} or complex analytic subvariety of codimension 1 (the zero set of a real or complex analytic function) is called a
{\em singular hypersurface}. That is the definition is the same as above, but
we do not require $\operatorname{grad} \rho \not= 0$. Note that some authors leave out the word {\em singular} and then use {\em non-singular hypersurface} for a hypersurface which is also a manifold. Some authors use the word {\em hypervariety} to describe a singular hypersurface.
An example of a hypersurface is the hypersphere (of radius 1 for simplicity) which has the defining equation
\begin{equation*}
x_1^2 + x_2^2 + \ldots + x_n^2 = 1 .
\end{equation*}
Another example of a hypersurface would be the boundary of a domain in
${\mathbb{C}}^n$ with smooth boundary.
An example of a singular hypersurface in ${\mathbb{R}}^2$ is for example the zero set
of $\rho(x_1,x_2) = x_1 x_2$ which is really just the two axis. Note that this
hypersurface fails to be a manifold at the origin.
%FIXME: terrible reference (too specific)
\begin{thebibliography}{9}
\bibitem{ber:submanifold}
M.\@ Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
{\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}},
Princeton University Press,
Princeton, New Jersey, 1999.
\end{thebibliography}