Families of Vector Fields which Generate the Group of Diffeomorphisms

Given a compact manifold M , we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group DiffM . Let M be a smooth n-dimensional compact manifold, VecM the space of smooth vector fields on M and Diff0M the group of isotopic to the identity diffeomorphisms of M . Given f ∈ VecM , we denote by t 7→ e , t ∈ R, the flow on M generated by f ; then e , t ∈ R, is a one-parametric subgroup of Diff0M . Let F ⊂ VecM ; the subgroup of Diff0M generated by e tf , f ∈ F , t ∈ R, is denoted by GrF . Theorem. Let F ⊂ VecM ; if GrF acts transitively on M , then Gr {af : a ∈ C(M), f ∈ F} = Diff0M. Corollary 1. Let ∆ ⊂ TM be a completely nonholonomic vector distribution. Then any isotopic to the identity diffeomorphism of M has a form e1 ◦ · · · ◦ ek , where f1, . . . , fk are sections of ∆. Remark. Recall that Gr{f1, f2} acts transitively on M for a generic pair of smooth vector fields f1, f2. SISSA, Trieste & Steklov Math. Inst., Moscow SISSA, Trieste In this paper, smooth means C. 1 We start the proof of the theorem with an auxiliary lemma that is actually the main part of the proof. Let B ⊂ R be diffeomorphic to a cube, 0 ∈ B; we set C 0 (B) = {a ∈ C (B) : a(0) = 0} and assume that C 0 (B) is endowed with the standard C-topology. Lemma 1 (Main Lemma). Let Xi ∈ VecR , ai ∈ C (R), i = 1, . . . , n, and the following conditions hold: • span{X1(0), . . . , Xn(0)} = R , • ai(0) = 0, 〈d0ai, Xi(0)〉 < 0, i = 1, . . . , n; then there exist ǫ, ε > 0 and a neighborhood O of (ǫa1, . . . , ǫan) ∣