PlanetMath: algebraic manifold

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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.

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Other names:

algebraic submanifold, $k$

-algebraic manifold, $k$

-algebraic submanifold

Also defines:

Nash manifold, Nash submanifold

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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 4223 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)

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