PlanetMath: algebraic manifold

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (191)
Orphanage
Unclass'd (1)
Unproven (496)
Corrections (5)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

algebraic manifold

(Definition)

Definition 1 Let

be a field and let $M \subset k^n$ be a submanifold.

is said to be an algebraic manifold (or

-algebraic) if there exists an irreducible algebraic variety $V \subset k^n$ such that $\dim V = \dim M$ and $M \subset V$ . If $k = \mathbb{R}$ , then

is called a Nash manifold.

It can be proved that such a manifold is defined as the zero set of a finite collection of analytic algebraic functions.

Bibliography

1: M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.

"algebraic manifold" is owned by jirka.

(view preamble)

Other names:

algebraic submanifold, $k$

-algebraic manifold, $k$

-algebraic submanifold

Also defines:

Nash manifold, Nash submanifold

Log in to rate this entry.
(view current ratings)

Cross-references: analytic algebraic functions, collection, finite, zero set, manifold, variety, algebraic, irreducible, submanifold, field

This is version 3 of algebraic manifold, born on 2005-12-05, modified 2007-09-30.
Object id is 7518, canonical name is AlgebraicManifold.
Accessed 3988 times total.

Classification:

AMS MSC:	14-00 (Algebraic geometry :: General reference works )
	14P20 (Algebraic geometry :: Real algebraic and real analytic geometry :: Nash functions and manifolds)
	58A07 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Real-analytic and Nash manifolds)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact