local Nagano theorem

Theorem (Local Nagano Theorem).

Let ΩRn be an open neighbourhood of a point x0. Further let g be a Lie subalgebra of the Lie algebraMathworldPlanetmath of real analytic real vector fields on Ω which is also a Cω(Ω;R)-module. Then there exists a real analytic submanifold MΩ with x0M, such that for all xM we have


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

Here note that Tx(M) is the tangent space of M at x, Cω(Ω;) are the real analytic real valued functions on Ω. Also real analytic real vector fields on Ω are the real analytic sectionsPlanetmathPlanetmath of T(Ω), the real tangent bundleMathworldPlanetmath of Ω.


The germ of the manifoldMathworldPlanetmath M is called the local Nagano leaf of 𝔤 at x0.


The union of all connected real analytic embedded submanifolds of Ω whose germ at x0 coincides with the germ of M at x0 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 Ω.


  • 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