real analytic subvariety
Let be an open set.
Definition.
A closed set is called a real analytic subvariety of such that for each point , there exists a neigbourhood and a set of real analytic functions defined in , such that
If and all the are real polynomials, then is said to be a real algebraic subvariety.
If is not required to be closed, then it is said to be a local real analytic subvariety. Sometimes is called a real analytic set or real analytic variety. Similarly as for complex analytic sets we can also define the regular and singular points.
Definition.
A point is called a regular point if there is a neighbourhood of such that is a submanifold. Any other point is called a singular point.
The set of regular points of is denoted by or sometimes The set of singular points is no longer a subvariety as in the complex case, though it can be sown to be semianalytic. In general, real subvarieties is far worse behaved than their complex counterparts.
References
- 1 Jacek Bochnak, Michel Coste, Marie-Francoise Roy. . Springer, 1998.
Title | real analytic subvariety |
Canonical name | RealAnalyticSubvariety |
Date of creation | 2013-03-22 17:41:07 |
Last modified on | 2013-03-22 17:41:07 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 4 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 14P05 |
Classification | msc 14P15 |
Synonym | real analytic variety |
Synonym | real analytic set |
Related topic | SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic |
Defines | real algebraic variety |
Defines | real algebraic subvariety |
Defines | local real analytic subvariety |
Defines | regular point |
Defines | singular point |