Cartan’s umbrella

The Cartan’s umbrella refers to a certain class of examples of real analytic sets (in fact real algebraic usually) in 3, which are irreducible (not written as a union of proper subsetsMathworldPlanetmathPlanetmath that are also subvarietiesMathworldPlanetmath), and where there are regular points both of dimensionMathworldPlanetmath 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(x2+y2)-y3=0. Solving for z we get z=y3x2+y2. The graph of this function is shown in the following figure.

Figure 1: Graph of z=y3x2+y2

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.


  • 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