Families of vector fields which generate the group of diffeomorphisms
aa r X i v : . [ m a t h . DG ] A p r Families of Vector Fields which Generate theGroup of Diffeomorphisms
A. A. Agrachev ∗ M. Caponigro † Abstract
Given a compact manifold M , we prove that any bracket generat-ing and invariant under multiplication on smooth functions family ofvector fields on M generates the connected component of unit of thegroup Diff M . Let M be a smooth n -dimensional compact manifold, Vec M the space ofsmooth vector fields on M and Diff M the group of isotopic to the identitydiffeomorphisms of M .Given f ∈ Vec M , we denote by t e tf , t ∈ R , the flow on M generatedby f ; then e tf , t ∈ R , is a one-parametric subgroup of Diff M . Let F ⊂
Vec M ; the subgroup of Diff M generated by e tf , f ∈ F , t ∈ R , is denotedby Gr F . Theorem.
Let
F ⊂
Vec M ; if Gr F acts transitively on M , then Gr { af : a ∈ C ∞ ( M ) , f ∈ F } = Diff M. Corollary 1.
Let ∆ ⊂ T M be a completely nonholonomic vector distribu-tion. Then any isotopic to the identity diffeomorphism of M has a form e f ◦ · · · ◦ e f k , where f , . . . , f k are sections of ∆ . Remark.
Recall that Gr { f , f } acts transitively on M for a generic pair ofsmooth vector fields f , f . ∗ SISSA, Trieste & Steklov Math. Inst., Moscow † SISSA, Trieste In this paper, smooth means C ∞ .
1e start the proof of the theorem with an auxiliary lemma that is actuallythe main part of the proof. Let B ⊂ R n be diffeomorphic to a cube, 0 ∈ B ; weset C ∞ ( B ) = { a ∈ C ∞ ( B ) : a (0) = 0 } and assume that C ∞ ( B ) is endowedwith the standard C ∞ -topology. Lemma 1 (Main Lemma) . Let X i ∈ Vec R n , a i ∈ C ∞ ( R n ) , i = 1 , . . . , n ,and the following conditions hold: • span { X (0) , . . . , X n (0) } = R n , • a i (0) = 0 , h d a i , X i (0) i < , i = 1 , . . . , n ;then there exist ǫ, ε > and a neighborhood O of ( ǫa , . . . , ǫa n ) (cid:12)(cid:12) B ε in C ∞ ( B ε ) n such that the mapping Φ : ( b , . . . , b n ) (cid:0) e b X ◦ · · · ◦ e b n X n (cid:1)(cid:12)(cid:12) B ε (1) is an open map from O into C ∞ ( B ε ) n , where B ε = (cid:8) e s X ◦ · · · ◦ e s n X n (0) : | s i | ≤ ε, i = 1 , . . . , n (cid:9) . Sketch of proof.
Openness of the map (1) is derived from the Hamil-ton’s version of the Nash–Moser inverse function theorem [2]. Set ¯ a =( ǫa , . . . , ǫa n ). In order to apply the Nash–Moser theorem we have to in-vert the differential of Φ at ¯ a and show that inverse is “tame” with respectto ¯ a . Here we make computations only for fixed ¯ a and leave the boring checkof the tame dependence on ¯ a for the detailed paper.Note that e ǫa j X j are closed to identity diffeomorphisms, hence ∂ Φ ∂b i (cid:12)(cid:12) ¯ a isobtained from ∂∂b i e b i X i (cid:12)(cid:12) ǫa i by a closed to identity change of variables. Wehave (cid:18) ∂∂a e aX (cid:19) : b e aX ∗ (cid:18)Z e R t h da,X i◦ e τaX dτ b ◦ e taX dtX (cid:19) ◦ e aX . This equality follows from the standard “variations formula” (see [1]) andthe relation: (cid:0) e taX (cid:1) ∗ : X (cid:16) e R t h da,X i◦ e − τaX dτ (cid:17) X. Let us define an operator A ( a, X ) : C ∞ ( ˆ B ε ) → C ∞ (cid:16) ˆ B ε (cid:17) by the formula A ( a, X ) b = Z e R t h da,X i◦ e τaX dτ b ◦ e taX dt, B ε = (cid:8) e sX ( x ) : | s | ≤ ε, x ∈ Π n − (cid:9) and Π n − is a transversal to X small ( n − A ( εa i , X i ) , i =1 , . . . , n, implies invertibility of D ¯ a Φ.Now set X = { bX : b ∈ C ∞ ( M ) } ⊂ Vec M . The map( bX ) ( A ( a, x ) b ) X has a clear intrinsic meaning as a linear operator on the space X ; moreover,this operator depends only on the vector field aX ∈ X . Indeed,( A ( a, X ) b ) X = e − aX ∗ (cid:0) D ( aX ) Exp (cid:12)(cid:12) X ( bX ) (cid:1) ◦ e − aX , where D Y Exp is the differential at the point Y ∈ Vec M of the map Exp : Y e Y , Y ∈ Vec M. Recall that a (0) = 0 , h d a, X (0) i <
0. In particular, X is transversal tothe hypersurface a − (0). We may rectify the field X in such a way that, innew coordinates, X = ∂∂x , a (0 , x , . . . , x n ) = 0. Now the field aX can betreated as a depending on y = ( x , . . . , x n ) family of 1-dimensional vectorfields a ( x , y ) ∂∂x . Moreover, a (0 , y ) = 0 , ∂a∂x (0 , y ) = α ( y ) < . A hyperbolic 1-dimensional field a ( x , y ) ∂∂x can be linearized by a smoothchange of variable and this smooth change of variable smoothly depends on y . Hence we may assume that aX = α ( y ) x ∂∂x . Then b ◦ e taX ( x , y ) = b ( e α ( y ) t x , y ).We thus have to invert the operatorˆ A : b ( x , y ) Z e − tα ( y ) b (cid:0) e α ( y ) t x , y (cid:1) dt acting in the space of smooth functions on a box. We can write b ( x , y ) = b ( y ) + x b ( y ) + x u ( x , y ) , where u is a smooth function. Then ˆ Ab = α (1 − e − α ) b , ˆ A ( x b ) = x b andˆ A (cid:0) x u ( x , y ) (cid:1) = x
21 1 Z e α ( y ) t u (cid:0) e α ( y ) t x , y (cid:1) dt = − x α ( y ) Z e α ( y ) u ( τ x , y ) dτ. B : u ( x , y ) Z e α ( y ) u ( τ x , y ) dτ. We set v ( x , y ) = x R x u ( s, y ) ds ; then( Bu )( x , y ) = (cid:0) v ( x , y ) − e α ( y ) v ( e α ( y ) x , y ) (cid:1) . (2)We introduce one more operator: R : v ( x , y ) e α ( y ) v ( e α ( y ) x , y ) . Let k v k C k, = sup ≤ i ≤ k (cid:13)(cid:13) ∂ i v∂x i (cid:13)(cid:13) C . Obviously, k R k C k, ≤ e sup α < , ∀ k . Hence( I − R ) − transforms a smooth on the box function ψ in the function ϕ =( I − R ) − ψ that is smooth with respect to x . As usually, the chain rule forthe differentiation allows to demonstrate that function ϕ is also smooth onthe box and to compute its derivatives: ∂ϕ∂y i = ( I − R ) − (cid:18) ∂ψ∂y i − e α ∂α∂y i ϕ − e α ∂α∂y i ∂ϕ∂x (cid:19) , e . t . c . Coming back to equation (2), we obtain: v = ( I − R ) − Bu . Finally, B − : w ∂∂x (cid:0) x ( I − R ) − w (cid:1) . (cid:3) Now set P = Gr { af : a ∈ C ∞ ( M ) , f ∈ F } , P q = { P ∈ P : P ( q ) = q } , q ∈ M. Lemma 2.
Any q ∈ M possesses a neighborhood U q ⊂ M such that the set n P (cid:12)(cid:12) U q : P ∈ P q o (3) has a nonempty interior in C ∞ q ( U q , M ) , where C ∞ q ( U q , M ) is the Fr´echetmanifold of smooth maps F : U q → M such that F ( q ) = q . roof. According to the Orbit Theorem of Sussmann [4] (see also thetextbook [1]), transitivity of the action of Gr F on M implies that T q M = span { P ∗ f ( q ) : p ∈ Gr F , f ∈ F } . Take X i = P i ∗ f i , i = 1 , . . . , n, such that P i ∈ Gr F , f i ∈ F , and X ( q ) , . . . , X n ( q )form a basis of T q M . Then for any vanished at q smooth functions a , . . . , a n ,the diffeomorphism e a X ◦ · · · ◦ e a n X n = P ◦ e ( a ◦ P ) f ◦ P − ◦ · · · ◦ P n ◦ e ( a n ◦ P n ) f n ◦ P − n belongs to the group P q . The desired result now follows from Main Lemma. Corollary 2.
Interior of the set (3) contains the identical map.
Proof.
Let O be an open subset of C ∞ q ( U q , M ) that is contained in (3)and P (cid:12)(cid:12) U q ∈ O . Then P − ◦ O is a contained in (3) neighborhood of theidentity. Definition 1.
Given P ∈ Diff M , we set supp P = { x ∈ M : P ( x ) = x } . Lemma 3.
Let O be a neighborhood of the identity in Diff M . Then for any q ∈ M and any neighborhood U q ⊂ M of q , we have: q ∈ int { P ( q ) : P ∈ O ∩ P , supp P ⊂ U q } . Proof.
Let vector fields X , . . . , X n be as in the proof of Lemma 2 and b ∈ C ∞ ( M ) a cut-off function such that supp b ⊂ U q and q ∈ int b − (1).Then the diffeomorphism Q ( s , . . . , s n ) = e s bX ◦ · · · ◦ e s n bX n belongs to O ∩ P for all sufficiently close to 0 real numbers s , . . . , s n and supp Q ( s , . . . , s n ) ⊂ U q . On the other hand, the map( s , . . . , s n ) Q ( s , . . . , s n )( q )is a local diffeomorphism in a neighborhood of 0. Lemma 4.
Let S j U j = M be a covering of M by open subsets and O be aneighborhood of identity in Diff M . Then the group Diff M is generated bythe subset { P ∈ O : ∃ j such that supp P ⊂ U j } . roof. The group Diff M is obviously generated by any neighborhoodof the identity. We may assume that the covering of M is finite and any U j is contained in a coordinate neighborhood. Moreover, taking a finer coveringand a smaller neighborhood O if necessary, we may assume that for any P ∈ O and any U j , the coordinate representation of P (cid:12)(cid:12) U j has a form P : x x + ϕ P ( x ), where ϕ is a C -small smooth vector function.Now consider a refined covering S i O i = M , so that O i ⊂ U j i for some j i and cut-off functions a i such that a i | O i = 1 , supp a i ⊂ U j i . Given P ∈ O , weset P i ( x ) = x + a i ( x ) ϕ P ( x ) , ∀ x ∈ U j i and P i ( q ) = q, ∀ q ∈ M \ U j i . Then supp ( P − i ◦ P ) ⊂ supp P \ O i . Now, by the induction with respect to i , we step by step arrive to a diffeomorphism with empty support. In otherwords, we present P as a composition of diffeomorphisms whose supports arecontained in U j . Proof of the Theorem.
According to Lemma 4, it is sufficient to provethat there exist a neighborhood U q ⊂ M and a neighborhood of the identity O ⊂
Diff M such that any diffeomorphism P ∈ O whose support is containedin U q belongs to P . Moreover, Lemma 3 allows to assume that P ( q ) = q .Finally, the corollary to Lemma 2 completes the job. Acknowledgment.
First coauthor is greatful to Boris Khesin who asked himthe question answered by this paper (see also recent preprint [3]).
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