# 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 LocalNaganoTheorem 2013-03-22 14:48:27 2013-03-22 14:48:27 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 17B99 msc 53B25 Nagano’s theorem local Nagano leaf Nagano leaf global Nagano leaf