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# 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 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. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.

## Mathematics Subject Classification

32V40*no label found*14J70

*no label found*

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