PlanetMath: local Nagano theorem

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local Nagano theorem

(Theorem)

Theorem 1 (Local Nagano Theorem) Let $\Omega \subset {\mathbb{R}}^n$ be an open neighbourhood of a point . Further let $\mathfrak{g}$ be a Lie subalgebra of the Lie algebra of real analytic real vector fields on $\Omega$ which is also a $C^\omega(\Omega;{\mathbb{R}})$ -module. Then there exists a real analytic submanifold $M \subset \Omega$ with $x^0 \in M$ , such that for all $x \in M$ we have

$\displaystyle T_x(M) = \mathfrak{g}(x) .$

Furthermore the germ of at is the unique germ of a submanifold with this property.

Here note that is the tangent space of at , $C^\omega(\Omega;{\mathbb{R}})$ 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

is called the local Nagano leaf of $\mathfrak{g}$ at

Definition 2 The union of all connected real analytic embedded submanifolds of $\Omega$ whose germ at

coincides with the germ of

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 $\Omega$ .

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.

"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 3178 times total.

Classification:

AMS MSC:	53B25 (Differential geometry :: Local differential geometry :: Local submanifolds)
	17B99 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Miscellaneous)

Pending Errata and Addenda

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