Cartan’s umbrella

The Cartan’s umbrella refers to a certain class of examples of real analytic sets (in fact real algebraic usually) in ${ℝ}^3$, 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.

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