# 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 $\rho :U\to \mathbb{R}$ with $\mathrm{grad}\rho \ne 0$ 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 ${\u2102}^{n}$ and we have a
hypersurface there it is called a real hypersurface in
${\u2102}^{n}$. $\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 $\mathrm{grad}\rho \ne 0$. 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

$${x}_{1}^{2}+{x}_{2}^{2}+\mathrm{\dots}+{x}_{n}^{2}=1.$$ |

Another example of a hypersurface would be the boundary of a domain in ${\u2102}^{n}$ 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 |