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local Nagano theorem
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(Theorem)
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Here note that $T_x(M)$ is the tangent space of $M$ at $x$ , $C^\omega(\Omega;{\mathbb{R}})$ are the real analytic real valued functions on $\Omega$ . Also real analytic real vector fields on $\Omega$ are the real analytic sections of $T(\Omega)$ , the real tangent bundle of $\Omega$ .
Definition 1 The germ of the manifold $M$ is called the local Nagano leaf of $\mathfrak{g}$ at $x_0$ .
Definition 2 The union of all connected real analytic embedded submanifolds of $\Omega$ whose germ at $x_0$ coincides with the germ of $M$ at $x_0$ is called the global Nagano leaf.
The global Nagano leaf turns out to be a connected immersed real analytic submanifold which may however not be an embedded submanifold of $\Omega$ .
- 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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"local Nagano theorem" is owned by jirka.
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| Other names: |
Nagano's theorem |
| Also defines: |
local Nagano leaf, Nagano leaf, global Nagano leaf |
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Cross-references: embedded submanifolds, connected, union, manifold, tangent bundle, sections, functions, tangent space, property, germ of a submanifold, germ, real analytic submanifold, vector fields, real, real analytic, Lie algebra, subalgebra, point, neighbourhood, open
This is version 3 of local Nagano theorem, born on 2004-11-10, modified 2005-03-07.
Object id is 6464, canonical name is LocalNaganoTheorem.
Accessed 4528 times total.
Classification:
| AMS MSC: | 53B25 (Differential geometry :: Local differential geometry :: Local submanifolds) | | | 17B99 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Miscellaneous) |
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Pending Errata and Addenda
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