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 wherethere 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.

单词	CartansUmbrella
释义	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 wherethere 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.
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