# 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