hypersurface

Definition.

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

M\cap U=\{x\in U\mid\rho(x)=0\}.

Then $M$ is called a hypersurface.

If $\rho$ is in fact smooth then $M$ is a smooth hypersurface and similarly if $\rho$ is real analytic then $M$ is a real analytic hypersurface. If we identify ${\mathbb{R}}^{2n}$ with ${\mathbb{C}}^{n}$ and we have a hypersurface there it is called a real hypersurface in ${\mathbb{C}}^{n}$ . $\rho$ is usually called the 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 real (http://planetmath.org/RealAnalyticSubvariety) or complex analytic subvariety of codimension 1 (the zero set of a real or complex analytic function) is called a 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 singular and then use non-singular hypersurface for a hypersurface which is also a manifold. Some authors use the word hypervariety to describe a singular hypersurface.

An example of a hypersurface is the hypersphere (of radius 1 for simplicity) which has the defining equation

x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=1.

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.

References

1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title	hypersurface
Canonical name	Hypersurface
Date of creation	2013-03-22 14:32:56
Last modified on	2013-03-22 14:32:56
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	8
Author	jirka (4157)
Entry type	Definition
Classification	msc 32V40
Classification	msc 14J70
Related topic	Submanifold
Defines	smooth hypersurface
Defines	real analytic hypersurface
Defines	real hypersurface
Defines	local defining function
Defines	singular hypersurface
Defines	non-singular hypersurface
Defines	hypervariety