The Cartan’s umbrella refers to a certain class of examples of real analytic sets (in fact real algebraic usually) in , 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 Solving for we get . The graph of this function is shown in the following figure.
Figure 1: Graph of
The umbrella itself also includes the axis, since all points of the form 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 axis is not included.
This pathological behavior does not happen for complex analytic subvarieties.
- 1 Jacek Bochnak, Michel Coste, Marie-Francoise Roy. . Springer, 1998.
|Date of creation||2013-03-22 17:41:11|
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