# Cartan’s umbrella

The Cartan’s umbrella refers to a certain class of examples of real analytic sets (in fact real algebraic usually) in ${\mathbb{R}}^{3}$,
which are irreducible (not written as a union of proper subsets^{} that are also subvarieties^{}), and where
there are regular points both of dimension^{} 1 and of dimension 2. Sometimes higher dimensional examples with similar behavior are also called the same. A fairly common equation for a Cartan umbrella is $z({x}^{2}+{y}^{2})-{y}^{3}=0.$ Solving for $z$
we get $z=\frac{{y}^{3}}{{x}^{2}+{y}^{2}}$. The graph of this function is shown in the following figure.

Figure 1: Graph of $z=\frac{{y}^{3}}{{x}^{2}+{y}^{2}}$

The umbrella itself also includes the $z$ axis, since all points of the form $(0,0,z)$ satisfy the equation. . It is impossible to write an equation (real analytic or real algebraic) whose solution set contains the graph in Figure 1, and such that the $z$ axis is not included.

This pathological behavior does not happen for complex analytic subvarieties.

## References

- 1 Jacek Bochnak, Michel Coste, Marie-Francoise Roy. . Springer, 1998.

Title | Cartan’s umbrella |
---|---|

Canonical name | CartansUmbrella |

Date of creation | 2013-03-22 17:41:11 |

Last modified on | 2013-03-22 17:41:11 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Example |

Classification | msc 14P05 |

Classification | msc 14P15 |

Synonym | Cartan umbrella |