local Nagano theorem

Theorem (Local Nagano Theorem).

Let $\Omega\subset{\mathbb{R}}^{n}$ be an open neighbourhood of a point $x^{0}$ . 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

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

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

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.

The germ of the manifold $M$ is called the local Nagano leaf of $\mathfrak{g}$ at $x_{0}$ .

Definition.

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 $\Omega$ .

References

1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title	local Nagano theorem
Canonical name	LocalNaganoTheorem
Date of creation	2013-03-22 14:48:27
Last modified on	2013-03-22 14:48:27
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Theorem
Classification	msc 17B99
Classification	msc 53B25
Synonym	Nagano’s theorem
Defines	local Nagano leaf
Defines	Nagano leaf
Defines	global Nagano leaf