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Example text

18). Let us now use for each plaque the homeomorphisms pr1 ◦uα |(u−1 α (Vα × m−k (V × {y}) → V ⊂ R as charts, then we describe on M a new {y})) : u−1 α α α smooth manifold structure ME with finer topology which however has uncountably many connected components, and the identity on M induces a bijective immersion ME → M . The connected components of ME are called the leaves of the foliation. In order to check the rest of the assertions made in the theorem let us construct the unique leaf L through an arbitrary point x ∈ M : choose a plaque containing x and take the union with any plaque meeting the first one, and keep going.

Draft from April 18, 2007 Peter W. Michor, 32 Chapter I. 25 Proof. (1) =⇒ (2). Let X ∈ XE and let L be the leaf through x ∈ M , with i∗ X i : L → M the inclusion. TF li∗ X (−t,x) L = EF lX (−t,x) . ∗ This implies that (FlX t ) Y ∈ XE for any Y ∈ XE . (2) =⇒ (4). In fact (2) says that XE ⊂ aut(E). ∗ (4) =⇒ (3). We can choose W = aut(E) ∩ XE : for X, Y ∈ W we have (FlX t ) Y ∈ XE ; so W ⊂ S(W) ⊂ XE and E is spanned by W. (3) =⇒ (1). We have to show that each point x ∈ M is contained in some integral submanifold for the distribution E.

14) the flow of this vector field is Xk ∗ Xk Xk Y X1 X1 1 Fl((FlX t1 ◦ · · · ◦ Fltk ) Y, t) = Fl−tk ◦ · · · ◦ Fl−t1 ◦ Flt ◦ Flt1 ◦ · · · ◦ Fltk , so S(W) is the minimal stable set of local vector fields which contains W. Now let F be an arbitrary distribution. A local vector field X ∈ Xloc (M ) is called an infinitesimal automorphism of F , if Tx (FlX t )(Fx ) ⊂ FFlX (t,x) whenever defined. We denote by aut(F ) the set of all infinitesimal automorphisms of F . By arguments given just above, aut(F ) is stable.

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