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algebraic manifold
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(Definition)
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Definition 1 Let $k$ be a field and let $M \subset k^n$ be a submanifold. $M$ is said to be an algebraic manifold (or $k$ 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 $M$ 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.
- 1
- M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.
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"algebraic manifold" is owned by jirka.
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(view preamble | get metadata)
| Other names: |
algebraic submanifold,  -algebraic manifold, -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 6023 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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Pending Errata and Addenda
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