local Nagano theorem

Theorem (Local Nagano Theorem).

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

$$T_x(M)=𝔤(x).$$

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

Here note that $T_x(M)$ is the tangent space of $M$ at $x$,$C^{\omega }(\mathrm{\Omega };ℝ)$ are the real analytic real valued functionson $\mathrm{\Omega }$. Also real analytic real vector fields on $\mathrm{\Omega }$ are thereal analytic sections of $T(\mathrm{\Omega })$, the real tangent bundle of $\mathrm{\Omega }$.

Definition.

The germ of the manifold $M$ is called the local Nagano leaf of$𝔤$ at $x_0$.

Definition.

The union of all connected real analytic embedded submanifolds of $\mathrm{\Omega }$ whosegerm at $x_0$ coincides with the germ of $M$ at $x_0$ is called theglobal 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 $\mathrm{\Omega }$.