The Group of Contact Diffeomorphisms for Compact Contact Manifolds
TTHE GROUP OF CONTACT DIFFEOMORPHISMS FOR COMPACTCONTACT MANIFOLDS
J. BLAND AND T. DUCHAMP
Abstract.
For a compact contact manifold M n +1 , it is shown that the anisotropic Folland-Steinfunction spaces Γ s ( M ) , s ≥ (2 n + 4) form an algebra. The notion of anisotropic regularity isextended to define the space of Γ s -contact diffeomorphisms, which is shown to be a topologicalgroup under composition and a smooth Hilbert manifold. These results are used in a subsequentpaper to analyse the action of the group of contact diffeomorphisms on the space of CR structureson a compact, three dimensional manifold. Introduction
Contact manifolds arise naturally in complex and CR geometry. The boundary of a stronglypseudoconvex domain is a contact manifold, and more generally, any strongly pseudoconvex CRmanifold is a contact manifold. In each case, the ¯ ∂ b –operator, which may be thought to embody thetangential Cauchy Riemann equations, is a natural operator that arises in analysis. The associatedsecond order operator (cid:3) b is anisotropic, being second order in the holomorphic tangential directions,and only first order in the transverse directions. In [FS], Folland and Stein introduced someanisotropic function spaces, the anisotropic Sobolev spaces Γ s and the anisotropic Banach spacesΓ s,α to reflect this behaviour, and showed that these operators are solvable with good estimates inthese spaces.In recent years, much attention has been focused on the space of CR structures which a givencompact manifold admits. A theorem of Gray [G] states that all contact structures in the samehomotopy class are equivalent. It is natural, therefore, to fix an underlying contact structure andstudy the action of the space of contact diffeomorphisms on the space of CR structures that arecompatible with the fixed contact structure.In [CL], Cheng and Lee constructed a transverse slice for the action of the contact diffeomorphismgroup. They avoided using the anisotropic spaces in [CL] by working in the Nash Moser category.In [B], we restricted our attention to the case of the standard S ⊂ C , and used explicit informationto construct an anisotropic Hilbert space structure on contact diffeomorphisms near the identity.In this paper, we show how to generalize the construction in [B], and show that, for generalcompact contact manifolds, contact diffeomorphisms form a smooth Hilbert manifold modelled onthe anisotropic Hilbert spaces. In [BD3], we apply our results to construct transverse slices forthe action of the group of contact diffeomorphisms on the space of compatible Cauchy-Riemannstructures on three dimensional compact contact manifolds. Date : June 30, 2010.2000
Mathematics Subject Classification.
Key words and phrases.
Cauchy-Riemann structure, contact structure, contact diffeomorphism, Folland-Steinspace.The first author was partially supported by an NSERC grant. The second author was partially supported by anNSF grant. a r X i v : . [ m a t h . DG ] J u l J. BLAND AND T. DUCHAMP
While our main interest is in the study of the action of contact diffeomorphisms on the space ofcompatible CR structures, the space of contact diffeomorphisms is of independent interest. Smoothcontact diffeomorphisms were first studied by Gray [G]. Later, Omori [O1, O2] worked withinthe category of ordinary Sobolev spaces to show that the space of contact diffeomorphisms isan ILH (
Inverse Limit Hilbert ) Lie group. We note that, in the special case where the contactmanifold admits a free transverse S action, Biquard [Bi] used a different method to obtain a localparameterization for contact diffeomorphisms near the identity.In this paper, we study the group of contact diffeomorphisms from the perspective of Folland-Stein function spaces, the natural function spaces respecting the contact structure. As we showin [BD3], by working in Folland-Stein spaces, we are able to obtain normal form theorems for theaction of CR diffeomorphisms on CR structures with only finite regularity.The paper is structured as follows. In Section 3, we introduce the anisotropic function spaces,and show that for s ≥ (2 n + 4), they form an algebra. We believe that the sharp result herewould be that the intersection of Γ s with the space of bounded functions is an algebra for all s ;however, the result which we have stated is sufficient for our purposes. In Section 2.6, we reviewRumin’s complex, and state the results which we will use in the analysis of the space of contactdiffeomorphisms. In Section 4, we show that Ebin’s trick of constructing a local coordinate systemfor the space of diffeomorphisms works equally well for the anisotropic Folland-Stein spaces, andthen use Rumin’s estimates to show the space of contact diffeomorphisms locally forms a smoothHilbert submanifold within the coordinate chart. Using this result, we easily prove that the space ofcontact diffeomorphisms is a smooth Hilbert manifold modelled on the anisotropic function spaces.1.1. Notation.
We summarize here the notation and conventions used throughout the paper.If A is a subset of a topological space X , then A denotes the closure of A in X . If A and B aresubsets of X , then the notation A (cid:98) B means that A is compactly contained in B .If F : A → B is a map between Banach spaces, with norms (cid:107) · (cid:107) A and (cid:107) · (cid:107) B , respectively, thenthe expression (cid:107)F ( f ) (cid:107) B ≺ (cid:107) f (cid:107) A means that there is a constant C > (cid:107)F ( f ) (cid:107) B ≤ C (cid:107) f (cid:107) A for all f ∈ A .We give R m the standard inner product ( · , · ), and we let |·| denote the corresponding norm.The symbols ( · , · ) s and (cid:107) · (cid:107) s , for s = 0 , , . . . denote the Folland-Stein inner products and norms,respectively.If N is a smooth manifold, then T N and T ∗ N denote its tangent and cotangent bundles, respec-tively; Λ p N denotes the p -th exterior power of T ∗ N ; Ω p ( N ) denotes the space of smooth p -formson N ; L X β denotes the Lie derivative of the form β with respect to the vector field X ; and X β denotes interior evaluation. If N has a Riemannian metric, then | X | denotes the norm of thetangent vector X with respect to that metric.The symbol C r ( E ), r = 0 , , , . . . , ∞ , denotes the space of C r -sections of a fiber bundle E → M ,equipped with the topology of uniform convergence of derivatives up to order r on compact sets.Similarly, C r ( M, N ) denotes the space of C r maps from M to N We endow R n +1 with the contact structure defined by the one-form η = dx n +1 − n (cid:88) j =1 x n + j dx j , ONTACT DIFFEOMORPHISMS 3 where ( x , . . . , x n , x n +1 , . . . , x n , x n +1 ) are the standard coordinates on R n +1 , and we let dV denote the standard volume form: dV = 1 n ! η ∧ ( dη ) n . We denote the contact distribution of η by H ⊂ T R n +1 and we set T = ∂∂x n +1 , X j = ∂∂x j + x n + j ∂∂x n +1 , and X n + j = ∂∂x n + j , ≤ j ≤ n . Observe that the collection { X j , ≤ j ≤ n } is a global framing for H . Notice that the 1-forms η , dx j , dx n + j , ≤ j ≤ n, are the dual coframe to T , X j , X n + j , 1 ≤ j ≤ n .Let f = ( f , . . . , f m ) be a smooth, R m -valued function defined on the closure of a domain D (cid:98) R n +1 . We define X I f = (cid:40) X i X i . . . X i t f for t > f for t = 0 , where we have introduced the multi-index notation I = ( i , . . . , i t ), 1 ≤ i j ≤ n and X I f =( X I f , . . . , X I f m ). (For t = 0, I denotes the empty index I = ().) The integer t is called the order of I and written | I | .Throughout this paper, M denotes a fixed smooth, compact contact manifold of dimension 2 n +1,with contact distribution H ⊂ T M . We call sections of H horizontal vector fields . We let π H : T ∗ M → H ∗ denote the projection map; by abuse of notation, we also let π H : Λ p M → Λ p H ∗ denote the extension of π H to the exterior product bundles. For convenience, we assume that M supports a fixed contact one-form η . The characteristic (or Reeb ) vector field T is the uniquevector field satisfying the conditions T η = 1 and
T dη = 0. We can then identify the dualcontact distribution with the annihilator of T , i.e. H ∗ = { β ∈ T ∗ M : T β = 0 } ⊂ T ∗ M ;more generally Λ p H ∗ = { β ∈ Λ p ( M ) : T β = 0 } , and we have the identity(1.1.1) π H ( β ) = T ( η ∧ β ) . Two forms β and β on M are said to be equal mod η , written β = β mod η , if and only if β = β + η ∧ α , for some α ∈ Ω ∗ ( M ). An easy exercise in the exterior calculus provesthe equivalences β = β mod η ⇐⇒ η ∧ ( β − β ) = 0 ⇐⇒ T (( β − β ) ∧ η ) = 0 ⇐⇒ π H ( β ) = π H ( β ) . None of our results depend on this assumption, for if the line bundle
T M/H is non-trivial, we can lift to a doublecover of M , where a global contact form does exist. J. BLAND AND T. DUCHAMP
We will also fix an endomorphism J : H → H such that J = − Id and such that the operator X (cid:55)→ dη ( X, J X ) is non-negative. (Such an endomorphism always exists.) We let let g denote theRiemannian metric defined by the formula g ( X, Y ) = η ( X ) η ( Y ) + dη ( X, J Y ) , were we have extended J to a map J : T M → T M by setting J ( T ) = 0. The endomorphism J and the metric g are said to be adapted to the contact structure. Finally, ∗ denotes the Hodge staroperator associated to the metric g .We say that a chart φ : U → R n +1 for M is an adapted coordinate chart if η = φ ∗ η . It followsthat the identities φ ∗ T = T and φ ∗ H = H hold for φ adapted. An adapted atlas for M is a finite,smooth atlas { φ α : U α → R n +1 } , consisting of adapted coordinate charts, together with openregions D α (cid:98) φ α ( U α ) such that { W α = φ − α ( D α ) } covers M . By compactness of M and Darboux’sTheorem for contact structures [Arn, page 362], M has an adapted atlas. An adapted coordinatechart for a fibre bundle π : E → M with m -dimensional fibres is a coordinate chart for E of theform ψ : U → φ ( V ) × R m : q (cid:55)→ ( φ ( π ( q )) , χ ( q ))with ψ surjective and where φ : V → R n +1 is an adapted chart for M . The chart is said to be centered at the point q if in addition ψ ( q ) = (0 , σ : V → U is a local section of E , thefunction f σ : φ ( V ) → R m defined by the formula f σ = χ ◦ σ ◦ φ − is called the local representation of σ .2. Global analysis on contact manifolds
In this section, we develop some of the analytical machinery we need to study the space ofcontact diffeomorphisms of M . We begin by introducing the Folland-Stein spaces Γ s associated toa compact contact manifold. We then introduce the notions of horizontal jet of a section of a fibrebundle and define the notion of contact order of a differential operator. We close this section with adiscussion of Rumin’s Complex [R] and its associated Hodge theory, which we need in Section 4 toconstruct a local parameterization of the group of Γ s -contact diffeomorphisms of a compact contactmanifold M . Most of the results in this section are extensions of definitions and theorems in [Pal]and [E] to the context of contact manifolds.2.1. Folland-Stein function spaces.
Let D be a bounded domain in R n +1 . The Folland-Steinspace Γ s ( D, R m ) is the Hilbert space completion of the set of smooth, R m -valued functions on D (the closure of D ) with respect to the inner product( f, g ) D,s := (cid:88) ≤| I |≤ s (cid:90) D ( X I f X I g ) dV . The associated norm is written (cid:107) f (cid:107) D,s = (cid:112) ( f, f ) D,s . When no confusion is likely to arise, wesuppress reference to D and write (cid:107) f (cid:107) s ; and we set Γ s ( D ) := Γ s ( D, R ). Remark 2.1.1.
Although we used the contact framing { X j : 1 ≤ j ≤ n } of H and the volumeform dV = n ! η ∧ ( dη ) n to define the inner product, an equivalent norm results if the framingis replaced by any smooth framing of H and dV is replaced by any smooth volume form on ¯ D .In particular, suppose that D (cid:48) ⊂ R n +1 is another bounded domain and F : D → D (cid:48) is a smooth ONTACT DIFFEOMORPHISMS 5 diffeomorphism that restricts to a contact diffeomorphism between D and D (cid:48) . Then compositionwith F induces an isomorphism Γ F : (cid:40) Γ s ( D (cid:48) , R m ) → Γ s ( D, R m ) f (cid:55)→ f ◦ F between Banach spaces. To see this, notice that because the derivative of F respects the contactdistribution, X I ( f ◦ F ) is a linear combination of terms of the form c J · ( X J f ) ◦ F , | J | ≤ s , where c J denotes a smooth function formed from F and its derivatives. It follows that f ◦ F is of class Γ s .We caution the reader that the condition that F be a contact diffeomorphism is essential. For if F does not preserve the contact distribution then the expansion of X I ( f ◦ F ) will in general involveterms of the form X J f with | J | > s , which may not be square integrable.The next lemma follows immediately from the definition of Γ s . Lemma 2.1.2.
The estimate (cid:107) f (cid:107) s ≺ (cid:88) j (cid:107) X j f (cid:107) s − + (cid:107) f (cid:107) is satisfied for all f ∈ Γ s ( D ) , s > . We shall repeatedly make use of the following Sobolev Lemma for Folland-Stein spaces, which isan immediate corollary of [FS, Theorem 21.1].
Lemma 2.1.3.
Let D (cid:48) (cid:98) D (cid:98) R n +1 , and let s = k + n + 2 , k ≥ . Let f ∈ Γ s ( D, R m ) . Then thefunctions X I f are continuous on ¯ D (cid:48) for all multi-indices of order | I | ≤ k . Moreover, max x ∈ ¯ D (cid:48) | X I f ( x ) | ≺ (cid:107) f (cid:107) D,s for all | I | ≤ k . If (cid:107) f (cid:107) D,s < ∞ for s = 2 k + n + 2 , then f is of class C k on ¯ D (cid:48) . Moreover, the linearmap Γ s ( D, R m ) → C k ( D (cid:48) , R m ) defined by restriction to D (cid:48) is continuous. Estimates for algebraic operations.
In this section, we prove some basic estimates. Lem-mas 2.2.1 and 2.2.2 are needed for our proof in Section 3 that composition and inverses of contactdiffeomorphisms are continuous operations. Lemma 2.2.3 and Proposition 2.3.5 are fundamentalestimates used throughout the paper.
Lemma 2.2.1.
Let s ≥ n + 3 , k ≤ s and consider open sets D (cid:48) (cid:98) D (cid:98) R n +1 . Then, for anyfunctions f ∈ Γ s ( D ) , g ∈ Γ k ( D ) , (cid:107) f · g (cid:107) D (cid:48) ,k ≺ (cid:107) f (cid:107) D,s · (cid:107) g (cid:107) D,k . Consequently, multiplication extends to a smooth bilinear mapping Γ s ( D ) × Γ k ( D ) → Γ k ( D (cid:48) ) . Proof.
We need only prove the estimate for smooth functions f and g on D . Recall that (cid:107) f · g (cid:107) D (cid:48) ,k = (cid:88) | I |≤ k (cid:90) D (cid:48) | X I ( f · g ) | dV . J. BLAND AND T. DUCHAMP
Applying the Leibniz rule, we find that (cid:107) f · g (cid:107) D (cid:48) ,k ≺ (cid:88) | J | + | K |≤ k (cid:90) D (cid:48) | X J ( f ) | | X K ( g ) | dV , There are two cases to consider: k ≤ ( n + 1) and k > n + 1. In the first case | J | ≤ n + 1 forevery summand, and we have the estimates (cid:90) D (cid:48) | X J f | | X K g | dV ≺ (cid:18) sup x ∈ D (cid:48) | X J f ( x ) | (cid:19) (cid:107) X K g (cid:107) D (cid:48) , ≺(cid:107) f (cid:107) D, | J | + n +2 (cid:107) g (cid:107) D,k ≺(cid:107) f (cid:107) D,s (cid:107) g (cid:107) D,k , where we have used the Sobolev inequality (Lemma 2.1.3) at the penultimate inequality.In the latter case, in each term either | J | ≤ n + 1 or | K | ≤ k − ( n + 2). In the first instance webound the term by (cid:107) f (cid:107) D,s (cid:107) g (cid:107) D,k as before; in the latter case we have (cid:90) D (cid:48) | X J ( f ) | | X K ( g ) | dV ≺(cid:107) X J f (cid:107) D (cid:48) , sup x ∈ D (cid:48) | X K g ( x ) | ≺(cid:107) f (cid:107) D, | J | (cid:107) g (cid:107) D, | K | + n +2 ≺(cid:107) f (cid:107) D,s (cid:107) g (cid:107) D,k , where we have again made use of Lemma 2.1.3. Summing over all terms gives the final estimate. (cid:3) Lemma 2.2.2.
Let D (cid:48) and D be open sets with D (cid:48) (cid:98) D (cid:98) R n +1 , and let f be a function in Γ s ( D ) ,where s ≥ n + 3 . Suppose that /f is bounded from above on ¯ D (cid:48) by a positive constant C > .Then (cid:107) /f (cid:107) D (cid:48) ,s ≺ (1 + (cid:107) f (cid:107) D,s ) s . Consequently, /f is contained in Γ s ( D (cid:48) ) . Moreover if /f (cid:48) < C for another function f (cid:48) ∈ Γ s ( D ) then (cid:107) /f − /f (cid:48) (cid:107) D (cid:48) ,s ≺ (1 + (cid:107) f (cid:107) D,s ) s (1 + (cid:107) f (cid:48) (cid:107) D,s ) s (cid:107) f − f (cid:48) (cid:107) D,s . Proof.
We have to estimate the quantities (cid:90) D (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) X J (cid:18) f (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dV , for | J | = t ≤ s . Now, by the quotient and product rules each such term is bounded by a sum ofexpressions of the form (cid:90) D (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) X J f · X J f · . . . X J p ff p +1 (cid:12)(cid:12)(cid:12)(cid:12) dV , where | J | + | J | + · · · + | J p | = t . Notice that | J j | > s/ (cid:90) D (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ( X J f ) · ( X J f ) . . . ( X J p ) ff p +1 (cid:12)(cid:12)(cid:12)(cid:12) dV ≺ C p +1) · (cid:0) (cid:107) f (cid:107) D,s (cid:1) p − (cid:107) f (cid:107) D,s ≺ (1 + (cid:107) f (cid:107) D,s ) s Summing over all terms gives the first estimate. We note here that when | J | = n + 1, | J | + n + 2 = 2 n + 3; whence the condition s ≥ n + 3 in the statement ofthe Lemma. ONTACT DIFFEOMORPHISMS 7
The second estimate follows immediately by applying the first estimate and applying Lemma 2.2.1to the quantity 1 /f − /f (cid:48) = ( f (cid:48) − f ) /f f (cid:48) . (cid:3) A minor modification of the proof of Lemma 2.2.1 gives an estimate for the product of severalfunctions.
Lemma 2.2.3.
Let D (cid:48) (cid:98) D (cid:98) R n +1 , with D (cid:48) and D open and s ≥ n + 4 . Then (cid:107) f · · · f p (cid:107) D (cid:48) ,s ≺ (cid:107) f (cid:107) D,s . . . (cid:107) f p (cid:107) D,s for all f j ∈ Γ s ( D ) for j = 1 , , . . . , p .Moreover for s > n + 4 , (cid:107) f · · · f p (cid:107) D (cid:48) ,s ≺ p (cid:88) j =1 (cid:107) f (cid:107) D,s − . . . (cid:107) f j − (cid:107) D,s − (cid:107) f j (cid:107) D,s (cid:107) f j +1 (cid:107) D,s − . . . (cid:107) f p (cid:107) D,s − , for all f j ∈ Γ s ( D ) , j = 1 , , . . . , p .Proof. The proof is similar to the proof of Lemma 2.2.1. By the product rule, (cid:107) f . . . f p (cid:107) D (cid:48) ,s ≺ (cid:88) | J |≤ s (cid:90) D (cid:48) | X J ( f · · · f p ) | dV ≺ (cid:88) | J | + ··· + | J p |≤ s (cid:90) D (cid:48) | X J f | · · · | X J p f p | dV , where X I f is defined in Section 2.1. We need only bound each term in the right-hand summation.Since s ≥ n + 4, it follows that | J j | > s/ ≥ n + 2 for at most one multi-index, say J j in theright-hand sum and that (since n + 2 ≤ s/ | J i | + n + 2 ≤ s/ n + 2 ≤ s for i (cid:54) = j . Hence by Lemma 2.1.3, X J i f i is continuous and sup x ∈ D (cid:48) | X J i f i | ≺ (cid:107) f i (cid:107) D,s . Consequently, (cid:90) D (cid:48) | X J f | . . . | X J p f p | dV ≺ (cid:89) i (cid:54) = j sup x ∈ D (cid:48) | X J i f i | · (cid:90) D (cid:48) | X J j f j | dV ≺ (cid:107) f (cid:107) D,s . . . (cid:107) f p (cid:107) D,s , from which the first estimate follows.Now suppose that s > n + 4. Then in the previous paragraph | J i | + n + 2 ≤ s −
1; andsup x ∈ D (cid:48) | X J i f i | ≺ (cid:107) f i (cid:107) D,s − for i (cid:54) = j , yielding the estimate (cid:107) f · · · f p (cid:107) D (cid:48) ,s ≺ p (cid:88) j =1 (cid:107) f (cid:107) D,s − . . . (cid:107) f j − (cid:107) D,s − (cid:107) f j (cid:107) D,s (cid:107) f j +1 (cid:107) D,s − . . . (cid:107) f p (cid:107) D,s − ≺ p (cid:88) j =1 (cid:107) f (cid:107) D,s − . . . (cid:107) f j − (cid:107) D,s − (cid:107) f j (cid:107) D,s (cid:107) f j +1 (cid:107) D,s − . . . (cid:107) f p (cid:107) D,s − . (cid:3) The Folland-Stein space of sections of a vector bundle.
In this section we define theFolland-Stein of sections of a vector bundle over a contact manifold.We begin by extending the definition of the Folland-Stein space Γ s ( D, R m ) of functions to thespace Γ s ( M, R m ) of functions on a compact contact manifold. Let ( φ α , U α , D α ) be an adapted atlasfor M . A function f : M → R m is said to be a Γ s -function if the functions f α = f ◦ φ − α lie inΓ s ( D α , R m ) for all α . The formula ( f, g ) M,s = (cid:88) α ( f α , g α ) D α ,s J. BLAND AND T. DUCHAMP makes Γ s ( M, R m ) into a separable Hilbert space. The Sobolev Lemma 2.1.3 clearly extends to thissetting: Lemma 2.3.1.
Let s = k + n + 2 , and let Y j , j = 1 , , . . . k be smooth sections of H ⊂ T M . Thenfor any function f ∈ Γ s ( M, R m ) , the functions Y Y . . . Y k f are continuous on M . Moreover, max x ∈ M | Y Y . . . Y k f ( x ) | ≺ (cid:107) f (cid:107) M,s If (cid:107) f (cid:107) M,s < ∞ for s = 2 k + n + 2 , then f is of class C k on M . In particular for s = 2 k + n + 2 ,the linear map Γ s ( M, R m ) → C k ( M, R m ) is continuous. Similarly, Lemma 2.2.3 assumes the following global form:
Lemma 2.3.2. If s ≥ n + 4 (cid:107) f · · · f p (cid:107) M,s ≺ (cid:107) f (cid:107) M,s . . . (cid:107) f p (cid:107) M,s , and if s > n + 4 , (cid:107) f · · · f p (cid:107) M,s ≺ k (cid:88) j =1 (cid:107) f (cid:107) M,s − . . . (cid:107) f j − (cid:107) M,s − (cid:107) f j (cid:107) M,s (cid:107) f j +1 (cid:107) M,s − . . . (cid:107) f p (cid:107) M,s − , for all f j ∈ Γ s ( M ) , j = 1 , , . . . , p . We define the Hilbert space of
Folland-Stein sections Γ s ( E ) for π : E → M a smooth vectorbundle of rank m as follows. View Γ s ( M, R r ), r ≥
1, as the Folland-Stein space of sections ofthe trivial vector bundle M × R r → M . For r sufficiently large, there is a vector bundle injection E ι (cid:44) → M × R r . Define an inner product on C ∞ ( E ) by the formula( f, g ) E,s = ( f ◦ ι, g ◦ ι ) M,s for f, g ∈ C ∞ ( E ), and let Γ s ( E ) be the Hilbert space completion of C ∞ ( E ) with respect to thisinner product. It is not difficult to check that, although the inner product depends on ι , the spaceΓ s ( E ) does not. Remark 2.3.3.
Because Γ s ( E ) is a closed subset of Γ s ( M, R r ), the Sobolev Lemma (2.3.1) extendsto this setting,The next proposition is the analogue of “Axiom B2” of Palais (see [Pal, page 10]) in the settingof contact manifolds. Proposition 2.3.4.
Let F : M → N be a smooth contact diffeomorphism between two compact,contact manifolds. Let E → N be a smooth vector bundle over N and let F ∗ E → M be its pull-backto M . Then the map σ (cid:55)→ σ ◦ F is a Hilbert space isomorphism between Γ s ( E ) and Γ s ( F ∗ E ) .Proof. Choose a bundle injection ι : E (cid:44) → N × R r , and let { φ α : U α → R n +1 } be an adapted atlasfor N (see Introduction). Set U (cid:48) α = F − ( U α ) and φ (cid:48) α = φ α ◦ F . Then { φ (cid:48) α : U (cid:48) α → R n +1 } is anadapted atlas for M . Use this atlas to define the inner product on Γ s ( M, R r ). Then by construction( f ◦ F, g ◦ F ) s,M = ( f, g ) s,N for all f, g ∈ Γ s ( N, R r ). Restricting f and g to sections of E then gives the result. (cid:3) The next proposition shows that Γ s satisfies “Axiom B5” of Palais ([Pal, page 39] for all s ≥ n +4. ONTACT DIFFEOMORPHISMS 9
Proposition 2.3.5.
Let E j → M , j = 1 , be smooth vector bundles over M and let F : E → E be a smooth (not necessarily linear) fibre-preserving map. Then the map Γ F : Γ s ( E ) → Γ s ( E ) : σ (cid:55)→ F ◦ σ is a C ∞ map for all s ≥ n + 4 .Moreover, if E is equipped with norm | · | E , then for every σ ∈ Γ s ( E ) and every c > there isa polynomial Q with non-negative coefficients of total degree at most s such that (*) (cid:107) F ◦ σ − F ◦ σ (cid:48) (cid:107) s ≤ (cid:107) σ − σ (cid:48) (cid:107) s Q s ( (cid:107) σ (cid:107) s , (cid:107) σ (cid:48) (cid:107) s ) for all σ (cid:48) ∈ Γ s ( E ) , with | σ (cid:48) − σ | < c .Proof. Our proof follows a similar argument in [Pal, Theorem 11.3]. We first show that Γ F satisfiesthe polynomial estimate (*), from which continuity of Γ F follows. It suffices to work in localcoordinates of an adapted atlas for M . Then Γ s sections of E j can be identified with elementsof Γ s ( D, R m j ), where m j is the fibre dimension of E j . Let f : D → R m be the local coordinaterepresentation of σ , and choose a constant c >
0. Choose σ (cid:48) so that it’s local representative g ∈ Γ s ( D, R m ) satisfies sup x ∈ D | g ( x ) − f ( x ) | < c . Then in local coordinates F ◦ σ ( x ) = F ( x, f ( x ))and F ◦ σ (cid:48) ( x ) = F ( x, g ( x )). By smoothness of F and compactness of D (cid:48) , all derivatives of F arebounded and smooth on the set { ( x, y ) : x ∈ D (cid:48) , | y − f ( x ) | < c . Hence, there is a fixed constant C > | F ( k ) ( x, y ) | < C and | F ( k ) ( x, y ) − F ( k ) ( x, z ) | < C | y − z | for all ( x, y ) and ( x, z ) with | y − f ( x ) | ≤ c and | z − g ( x ) | ≤ c , where F ( k ) ( x, y ) denotes any mixedpartial derivative of F of order k ≤ s .Next recall that (cid:107) F ◦ f − F ◦ g (cid:107) D (cid:48) ,s is a sum of integrals of the form(ii) (cid:90) D (cid:48) | X I { F ( x, f ( x )) − F ( x, g ( x )) } | dV for | I | ≤ s . By the chain rule, X I { F ( x, f ( x )) − F ( x, g ( x )) } is a finite sum of terms of the form(iii) F ( k ) ( x, f ( x )) · X I f ( x ) · · · X I k f ( x ) − F ( k ) ( x, g ( x )) · X I g ( x ) · · · X I k g ( x )= (cid:110) F ( k ) ( x, f ( x )) − F ( k ) ( x, g ( x )) (cid:111) · X I f ( x ) · · · X I k f ( x )+ ( F ( k ) ( x, g ( x )) · { X I f ( x ) · · · X I k f ( x ) − X I g ( x ) · · · X I k g ( x ) } where 0 ≤ k ≤ s and (cid:80) kj =1 | I j | ≤ s . Applying (i) to (iii) gives the estimate(iv) | F ( k ) ( x, f ( x )) · X I f ( x ) · · · X I k f ( x ) − F ( k ) ( x, g ( x )) · X I g ( x ) · · · X I k g ( x ) |≤ C | f ( x ) − g ( x ) | · | X I f ( x ) · · · X I k f ( x ) | + C | X I f ( x ) · · · X I k f ( x ) − X I g ( x ) · · · X I k g ( x ) | The right-hand side of (iv) can, in turn, be bounded by a finite sum of terms form(v) C | X I ( f ( x ) − g ( x )) | · | X I f ( x ) | . . . | X I k (cid:48) f ( x ) | · | X I k (cid:48) +1 g ( x ) | . . . | X I k (cid:48) + k (cid:48)(cid:48) g ( x ) | where 0 ≤ | I i | and (cid:80) k (cid:48) + k (cid:48)(cid:48) i =0 | I i | ≤ s . Substituting (v) into (ii) shows that (cid:107) F ◦ f − F ◦ g (cid:107) D (cid:48) ,s isbounded by a sum of integrals of the form(vi) (cid:90) D C | X I ( f ( x ) − g ( x )) | · | X I f ( x ) | . . . | X I k (cid:48) f ( x ) | · | X I k (cid:48) +1 g ( x ) | . . . | X I k (cid:48) + k (cid:48)(cid:48) g ( x ) || X J ( f ( x ) − g ( x )) | · | X J f ( x ) | . . . | X J (cid:96) (cid:48) f ( x ) | · | X J (cid:96) (cid:48) +1 g ( x ) | . . . | X J (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) g ( x ) | dV
00 J. BLAND AND T. DUCHAMP where 0 ≤ | J j | and (cid:80) (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) j =0 | J j | ≤ s .Notice that | I i | ≥ s/ | J j | ≥ s/ i and at most one j , and since s ≥ n +4, theSobolev Lemma 2.1.3 applies, to show that the remaining factors in the integrand are all continuous,hence bounded on the compact set { ( x, y ) : x ∈ D (cid:48) , | y − f ( x ) | ≤ c } . It follows that the integral in(vi) is bounded by an expression of the form C (cid:48) (cid:107) f − g (cid:107) D,s (cid:107) f (cid:107) k (cid:48) D,s (cid:107) g (cid:107) k (cid:48)(cid:48) D,s (cid:107) f (cid:107) (cid:96) (cid:48) D,s (cid:107) g (cid:107) (cid:96) (cid:48)(cid:48) D,s , for C (cid:48) a constant depending on s , F , c , and D (cid:48) . Since k (cid:48) + k (cid:48)(cid:48) ≤ s and (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) ≤ s , it follows that (cid:107) F ◦ f − F ◦ g (cid:107) D (cid:48) ,s ≺ (cid:107) f − g (cid:107) D,s Q ( (cid:107) f (cid:107) D,s , (cid:107) g (cid:107) D,s ) , where Q ( u, v ) is a polynomial of bidegree at most s in u and v , with non-negative coefficients.Applying this to each chart in an adapted atlas yields the global estimate (*).We now show that Γ F is C with derivative given by the formula d Γ F = Γ δF : Γ s ( E ) × Γ s ( E ) → Γ s ( E )where δF : E × M E → E is the smooth fibre bundle map defined by the formula δF x ( u, v ) = ddh (cid:12)(cid:12)(cid:12)(cid:12) h =0 F ( u + hv ) , for all x ∈ M and u, v ∈ E ,x .To show that lim v → (cid:107) Γ F ( σ + v ) − Γ F ( σ ) − Γ δF ( σ, v ) (cid:107) s (cid:107) v (cid:107) s = 0 , first observe that δF can be expressed as a smooth map of the form δF : E → Hom( E , E ) . Hence, Γ( δF ) : Γ s ( E ) → Γ s (Hom( E , E ))is continuous, and for all (cid:15) > δ such that (cid:107) Γ( δF )( σ + v ) − Γ( δF )( σ ) (cid:107) s < (cid:15) whenever (cid:107) v (cid:107) s < δ .Using this observation, we compute as follows:Γ F ( σ + v ) − Γ F ( σ ) − Γ δF ( σ, v ) = F ( σ + v ) − F ( σ ) − δF σ · v = (cid:90) { δF σ + tv · v − δF σ · v } dt = (cid:90) { Γ δF ( σ + tv )( v ) − Γ δF ( σ )( v ) } dt Hence, if (cid:107) v (cid:107) s < δ then (cid:107) Γ F ( σ + v ) − Γ F ( σ ) − Γ δF ( σ, v ) (cid:107) s ≤ (cid:90) (cid:107) Γ δF ( σ + tv )( v ) − Γ δF ( σ )( v ) (cid:107) s dt < (cid:15) (cid:107) v (cid:107) s . This show that Γ F is differentiable at σ . That it is continuously differentiable follows from theidentity d Γ F = Γ δF and continuity of Γ δF .That Γ F is smooth follows by induction. For assume that for some k >
0, Γ F is C k , for allsmooth F : E → E , and all E j . To show that Γ F is C k +1 , we need only show that its derivative d Γ F : Γ s ( E × E ) → Γ s ( E ) is C k . But d Γ F = Γ δF , and δF is, a smooth fibre bundle map.Consequently, d Γ F is C k , completing the induction step. (cid:3) ONTACT DIFFEOMORPHISMS 11
The Folland-Stein space of sections of a fibre bundle.
In this section we define theFolland-Stein space Γ s ( E ) of sections of E for s ≥ n + 4 in the case where π : E → M is a smoothfibre bundle over M . For this range of s , Γ s ( E ) is a smooth infinite dimensional manifold modelledon the Folland-Stein space of sections of certain vector bundles. The construction and the proofare due to Palais (see [Pal, Chapters 12 and 13] for details). We emphasize that this constructiondepends heavily on Propositions 2.3.4 and 2.3.5 (Palais’ Axioms B2 and B5), which are satisfiedfor s ≥ n + 4.Let π : E → M be a fibre bundle, and choose a smooth section σ ∈ C ∞ ( E ). The bundle ofvertical tangent vectors along σ is the vector bundle defined by T σ ( E ) = { X ∈ T E σ ( x ) : x ∈ M, dπ ( X ) = 0 } . Palais shows that there is a smooth fibre bundle isomorphism(2.4.1) ψ σ : T σ ( E ) (cid:39) −→ O σ open ⊂ E where O σ is a neighbourhood of the image of σ . Palais also shows that the the image of everycontinuous section of E is contained in a set of the form O σ for some smooth section σ .Consequently every continuous section of E can be identified with a continuous section of T σ ( E )for some σ ∈ C ∞ ( E ), and C ( E ) can we written as the following union of open sets: C ( E ) = (cid:91) σ ∈ C ∞ ( E ) C ( T σ ( E )) . Since s ≥ n +4 ≥ n +2, Lemma 2.3.1 applies to give continuous inclusions Γ s ( T σ ( E )) ⊂ C ( T σ ( E ).We may thus define the Folland-Stein space Γ s ( E ) to be the unionΓ s ( E ) = (cid:91) σ ∈ C ∞ ( E ) Γ s ( T σ ( E )) , equipped with the weakest topology such that Γ ψ σ : Γ s ( T σ ( E )) → Γ s ( E ) is a continuous open mapfor all σ ∈ C ∞ ( E ).In fact, Γ s ( E ) is a Hilbert manifold with C ∞ atlas given by the charts Γ( ψ − σ ); and as Palaisshows, smoothness of the transition functions for this atlas follows from Proposition 2.3.5.The above construction is functorial: Proposition 2.4.2 (Palais, Theorem 13.4) . Let E j → M , j = 1 , be smooth fibre bundles over M and let F : E → E be a smooth fibre-preserving map. Then the map Γ( F ) : Γ s ( E ) → Γ s ( E ) : σ (cid:55)→ F ◦ σ is a C ∞ map of Hilbert manifolds for all s ≥ n + 4 . Remark 2.4.3.
By construction, the Sobolev Lemma 2.3.1 extends to define a continuous injectionΓ s ( E ) ⊂ C k ( E ), for s = max(2 k + n + 2 , n + 4) .The case where E is the trivial fibre bundle E = M × N → M is an important special case: Definition 2.4.4.
Let N be a smooth manifold without boundary. For s ≥ n + 4, the Folland-Stein space Γ s ( M, N ) of maps from the contact manifold M to the manifold N is the Folland-Steinspace of sections of the trivial fibre bundle M × N → M . Corollary 2.4.5.
Let F : N → N (cid:48) be a C ∞ map between C ∞ manifolds. Left composition with F defines a C ∞ L sF : Γ s ( M, N ) → Γ s ( M, N (cid:48) ) : G (cid:55)→ F ◦ G for all s ≥ n + 4 . If F : N → N is a diffeomorphism of N then L sF is a diffeomorphism of Γ s ( M, N ) .Proof. View composition with F as a smooth bundle map ( x, y ) (cid:55)→ ( x, F ( y )), and apply Proposi-tion 2.4.2. If F is a diffeomorphism then R sF − is the smooth inverse of R sF . (cid:3) Horizontal jets and differential operators.
The goal of this section is to extend theframework of [Pal, Chapter 15] to the context of differential operators on contact manifolds.Let π : E → M be a smooth fibre bundle over M , and choose an adapted coordinate chart ψ : U → φ ( V ) × R m : p (cid:55)→ ( x, y ) , as defined in Section 1.1. Two smooth local sections σ i , i = 1 ,
2, of E defined on V , with localrepresentations f i : φ ( V ) → R m , are said to be contact equivalent up to order k at a point p ∈ V ifand only if(2.5.1) X I f ( φ ( p )) = X I f ( φ ( p ))for every multi-index I with 0 ≤ | I | ≤ k . It is easy to check that contact equivalence is anequivalence relation and that it is independent of coordinates. The horizontal k -jet of σ at p ,written j kH σ ( p ), is the equivalence class of the local section σ at p ∈ M , J kH E denotes the space ofall horizontal k -jets. The map π : J kH E → M defined by π (cid:16) j kH σ ( p ) (cid:17) = p makes the space of horizontal k -jets into a fibre bundle with fibres of dimension m · N k , where N k is the number of indices A with | A | ≤ k . Remark 2.5.2.
By virtue of the commutation relations among the vector fields T , X ,. . . X n , any differential operator of the form Y Y . . . Y r , where Y , . . . , Y r are arbitrary vector fields on an openset V ⊂ M , can be expressed uniquely in local coordinates as a linear combination of operators ofthe form D A := X · · · X (cid:124) (cid:123)(cid:122) (cid:125) a , X · · · X (cid:124) (cid:123)(cid:122) (cid:125) a , . . . , X n · · · X n (cid:124) (cid:123)(cid:122) (cid:125) a n T · · · T (cid:124) (cid:123)(cid:122) (cid:125) a n +1 where A = ( a , a , . . . , a n , a n +1 ), 0 ≤ a j . Moreover, if Y , . . . , Y r are all horizontal, then a + a + · · · + a n + 2 a n +1 = r . The integer | A | = a + · · · + a n + 2 a n +1 is called the contact order of D A .It follows that σ and σ are contact equivalent up to order k at p if and only if D A f ( φ ( p )) = D A f ( φ ( p ))for all multi-indices A with | A | ≤ k . Lemma 2.5.3.
Let E → M be a smooth fibre bundle with fibre dimension m . Then J kH E → M isa smooth fibre bundle. Moreover, if σ is a smooth section of E then j kH σ : p (cid:55)→ j kH σ ( p ) is a smoothsection of J kH E . If F : E → E is a smooth map of fibre bundles, then so is the map J kH ( F ) : J kH E → J kH E : j kH σ ( p ) (cid:55)→ j kH ( F ◦ σ )( p ) . This construction is functorial, i.e. J kH ( G ◦ F ) = J kH ( G ) ◦ J kH ( F ) , for G : E → E a smooth mapof fibre bundles. ONTACT DIFFEOMORPHISMS 13
Proof.
We give an outline of the proof, leaving some details to the reader. To define a coordinatechart for J kH E , choose a point q ∈ E and let p = π ( q ). Let ψ : U → φ ( V ) × R n : q (cid:55)→ ( x, y )be adapted coordinates for E centered at q (see Section 1.1); and let (cid:101) U be the set of horizontal k -jets of sections of E defined on V and with values in U ⊂ E . Viewing R m · N k as the space of m × N k matrices, we can define local fibre bundle coordinates (cid:101) ψ : (cid:101) U → R n +1 × R m · N k by the formula (cid:101) ψ ( j kH σ ( p )) = (( x ( p )) , ( D A f σ ( p ))) , where f σ is the local representation of σ (Section 1.1), D A f σ is the derivative of f σ with respectto the multi-index A , and we have given the set { A : | A | ≤ k } the lexicographical ordering.We need only show that (cid:101) ψ is a bijection between (cid:101) U and φ ( V ) × R m · N k for U a sufficiently smallneighbourhood of q . By the discussion in Remark 2.5.2, (cid:101) ψ is injective.To prove surjectivity, choose a point S = { S A ∈ R m : | A | ≤ k } ∈ R m · N k , and consider thepolynomial section y = σ S ( x ) = (cid:88) A A ! S A · ( x ) a . . . ( x n ) a n ( x n +1 ) a n +1 . where we have adopted the notation A ! = a ! . . . a n ! a n +1 !. Now for any point p = ( x , . . . , x n +1 ),the map S (cid:55)→ (cid:101) ψ ( σ S ( p )) = ( x, D A σ S ( x ))may be viewed as a linear map L x : R m · N k → R m · N k . Notice that L x depends smoothly on x . Toshow that L x is a bijection for x sufficiently near 0, we need only verify that L is injective. But astraightforward computation shows that D A (cid:48) σ S (0) = (cid:40) S A for A (cid:48) = A L is the identity map. This shows that (cid:101) ψ is surjective on the fibre of J kH E over all points p sufficiently near p . Thus, after a possible shrinking of U , the map (cid:101) ψ : (cid:101) U → R n +1 × R m · N k is a bijection between (cid:101) U and φ ( U ) × R m · N k .By letting q vary over all E , we obtain a smooth atlas for J kH E . We topologize J kH E by requiringeach of the charts (cid:101) ψ to be a homeomorphism. That these charts form a smooth atlas making J kH E into a smooth manifold follows from the observation that if (cid:101) ψ and (cid:101) ψ (cid:48) are two charts then (cid:101) ψ (cid:48) ◦ (cid:101) ψ − isa diffeomorphism between (cid:101) φ ( (cid:101) U ∩ (cid:101) U (cid:48) ) and (cid:101) φ (cid:48) ( (cid:101) U ∩ (cid:101) U (cid:48) ). The proof follows by standard arguments inadvanced calculus (i.e. the Inverse Function theorem and the Chain rule) and is left to the reader.That J kH ( F ) and j kH σ are smooth follows from the Chain Rule, as does the identity J kH ( G ◦ F ) = J kH ( G ) ◦ J kH ( F ). (cid:3) Remark 2.5.4.
In the special case where E → M is a smooth vector bundle, then so is J kH E → M ,with linear structure induced by the formula a j kH σ ( p ) + a j kH σ ( p ) = j kH ( a σ + a σ )( p ) . Lemma 2.5.5.
Let E → M be a smooth fibre bundle over M . Then the map Γ( j H ) : Γ s ( E ) → Γ s − k ( J kH E ) : σ (cid:55)→ j kH σ is a smooth map of Hilbert manifolds for all k and s such that s ≥ n + 4 + k . In the special casewhere E is a vector bundle, Γ( j H ) is a bounded linear map of Hilbert spaces.Proof. From the discussion in Section 2.1, it suffices to analyze Γ( j H ) in the neighbourhood ofa fixed section of E . By definition of the Hilbert manifold structure of Γ s ( E ) we may, withoutloss of generality assume that E is a vector bundle, and we must only show that the linear mapΓ( j H ) is bounded. By definition of the inner product on Γ s ( E ), the result then follows from a localcomputation: Choose an open set V (cid:98) M and a trivialization E | V (cid:39) V × R m . Then a section σ of E over V is given by an R m -valued function f σ and its horizontal k -jet j kH σ : V → J kH M can beidentified with the m × N k matrix-valued function j kH σ = ( D A f σ ) | A |≤ N k So (cid:107) j kH σ (cid:107) V,s − k = (cid:88) j (cid:88) ≤| J |≤ s − k (cid:90) V | X J ( j kH σ j ) | dV = (cid:88) ≤| J |≤ s − k (cid:88) | A |≤ k (cid:90) | X J D A f σ | dV ≤ (cid:88) ≤| I |≤ s (cid:90) | X I f σ | dV ≤ (cid:107) σ (cid:107) V,s . (cid:3) Definition 2.5.6.
Let E and E be smooth fibre bundles over M . A differential operator ofcontact order k from E to E is a map of the form D : C ∞ ( E ) j kH −→ C ∞ ( J kH E ) F ∗ −→ C ∞ ( E ) . where F : J kH E → E is a smooth fibre bundle map and F ∗ is defined by the formula F ∗ (ˆ σ ) = F ◦ ˆ σ for ˆ σ ∈ C ∞ ( J kH E ) . Proposition 2.5.7.
Let D be a differential operator of contact order k as above. Then D extendsto a smooth map D : Γ s ( E ) j kH −→ Γ s − k ( J kH E ) F ∗ −→ Γ s − k ( E ) , for all s ≥ n + 4 + k .In the special case where E and E are normed vector bundles, for every section σ ∈ Γ s ( E ) and every constant c > , there is a polynomial Q of degree at most s − k such that the estimate (cid:107) Dσ − Dσ (cid:48) (cid:107) s − k ≤ (cid:107) σ − σ (cid:48) (cid:107) s · Q ( (cid:107) σ (cid:107) s , (cid:107) σ (cid:48) (cid:107) s ) holds for every section σ (cid:48) ∈ Γ s ( E ) satisfying | j kH σ (cid:48) | ≤ c .Proof. The first part of the proposition is an immediate corollary to Propositions 2.5.5 and 2.3.5.The second part of the proposition follows from estimate in Proposition 2.3.5. (cid:3)
Remark 2.5.8.
The restriction s ≥ n + 4 + k in Proposition 2.5.7 can be relaxed to s ≥ k when D is a linear operator, and the estimate assumes the form (cid:107) Dσ (cid:107) s − k ≤ C (cid:107) σ (cid:107) s . When D is nonlinear,the term (cid:107) Dσ (cid:107) s − k involves products of σ and its derivatives, and estimating these expressions usesLemma 2.2.3, which assumes s − k ≥ n + 4. ONTACT DIFFEOMORPHISMS 15
Examples 2.5.9.
As examples, we consider the contact order of some basic operators that we needlater. Let M be a contact manifold with contact form η and characteristic vector field T .(i) Lie differentiation with respect to the Reeb vector field T L T : Γ s ( M ) → Γ s − ( M ) : f (cid:55)→ T ( f ) is a differential operator of contact order 2. To see this, we work locally, using anadapted coordinate chart φ : U → R n +1 as in Section 1.1. Note that φ ∗ T = T = ∂∂x n +1 .But since T = [ X , X n +1 ], the Lie bracket of the two horizontal vector fields, it followsthat T has contact order 2.(ii) The exterior derivative operator d : Γ s (Λ p M ) → Γ s − (Λ p +1 M ) : α (cid:55)→ dα is a differentialoperator of contact order 2. We again work locally. First consider the case p = 0, where α = f , for f a scalar function on M . In this case, df = n (cid:88) k =1 X k ( f ) dx k + T ( f ) η . Since the term T ( f ) η has contact order 2, and all other terms depend only on the horizontal1-jet of f , we see that dα has contact order 2. For p > α can be expressed in the form α = (cid:88) I f I dx I + (cid:88) K g K η ∧ dx K , where dx I = dx i ∧ · · · ∧ dx i p and η ∧ dx K = η ∧ dx k ∧ · · · ∧ dx k p − , where summationranges over indices I and K of the forms i < i < · · · < i p < n + 1 and k < k < · · · In [R], Rumin constructed a novel resolution0 (cid:44) → R −→ R d R −→ R d R −→ · · · d R −→ R n D R −→ R n +1 d R −→ R n +2 d R −→ · · · d R → R n +1 −→ n < p ≤ n + 1, R p denotes the subbundle of Λ p M given by R p := { β ∈ Λ p M : η ∧ β = 0 and dη ∧ β = 0 } and for 0 ≤ p ≤ n , R p denotes the quotient bundle R p = Λ p ( M ) /I p , where I = 0 and I p := { η ∧ α + dη ∧ β : α ∈ Λ p − M, β ∈ Λ p − M } for 0 < p ≤ n .Also note that for 0 ≤ p ≤ n , R p can be written as a quotient bundle of Λ p H ∗ :(2.6.1) τ : Λ p H ∗ → R p = Λ p H ∗ / ( dη ∧ Λ p − H ∗ ) . Let R p = C ∞ ( R p ). Then R = Ω ( M ), and since we make the identification H ∗ = R with theannihilator of T , R ≡ { α ∈ Ω ( M ) : T α = 0 } . The linear differential operators d R and D R are induced by the exterior derivative operator onforms. Let β ∈ R p be any section of R p There are three cases to consider:(i) For p > n , set d R β = dβ . It is easy to see that dβ is a section of R p +1 .(ii) For p < n , set d R β = π R ( d (cid:101) β ), where (cid:101) β ∈ Ω p ( M ) is any p -form with π R (cid:101) β = β and π R :Λ p M → R p denotes the quotient map. It is not difficult to check that d R β is independentof the choice of (cid:101) β .(iii) For p = n , set D R ( β ) = d (cid:101) β , where (cid:101) β ∈ Ω n ( M ) is an n -form satisfying the conditions π R (cid:101) β = β and d (cid:101) β ∈ R n +1 . Rumin shows that a form (cid:101) β satisfying these conditions existsand that d (cid:101) β is independent of the choice of (cid:101) β . ONTACT DIFFEOMORPHISMS 17 Rumin also shows that d R and D R are linear differential operators, with d R of contact order 1 and D R of contact order 2. Using the star operator, Rumin proves that R k is dual to R n +1 − k and thatthe adjoint operators satisfy the identities δ R = ( − k ∗ d R ∗ for k (cid:54) = ( n + 1) and D ∗ R = ( − n +1 ∗ D R ∗ . Thus δ R has contact order 1 and D ∗ R has contact order 2. The next proposition then follows fromProposition 2.5.7 above. Proposition 2.6.2 (Rumin) . Let ( M n +1 , η ) be a compact contact manifold with adapted metric g and the associated complex ( R ∗ , d R ) . Then the following estimates hold (cid:40) (cid:107) d R α (cid:107) s − ≤ c s (cid:107) α (cid:107) s , for k (cid:54) = n (cid:107) D R α (cid:107) s − ≤ c s (cid:107) α (cid:107) s , for k = n ; and (cid:40) (cid:107) δ R α (cid:107) s − ≤ c s (cid:107) α (cid:107) s , for k (cid:54) = n + 1 (cid:107) D ∗ R α (cid:107) s − ≤ c s (cid:107) α (cid:107) s , for k = n + 1 . Rumin[R] establishes a Hodge theory for this complex. The Laplace operators ∆ R : R k → R k are defined as follows(2.6.3) ∆ R = ( n − k ) d R δ R + ( n − k + 1) δ R d R , for 0 ≤ k ≤ ( n − , ( d R δ R ) + D ∗ R D R , for k = n,D R D ∗ R + ( δ R d R ) , for k = n + 1 , ( n − k + 1) d R δ R + ( n − k ) δ R d R , for ( n + 2) ≤ k ≤ (2 n + 1) . Theorem 2.6.4 (Rumin) . Let M be a contact manifold with adapted metric g . The Laplaceoperators ∆ R are maximal hypoelliptic, and the following estimates are satisfied for α ∈ R k : (cid:40) (cid:107) α (cid:107) s +2 ≤ c s (cid:107) ∆ R α (cid:107) s + (cid:107) α (cid:107) for k (cid:54) = n, ( n + 1) , (cid:107) α (cid:107) s +4 ≤ c s (cid:107) ∆ R α (cid:107) s + (cid:107) α (cid:107) for k = n, ( n + 1) . Remark 2.6.5. Rumin only establishes the estimates stated above for the case s = 0 (see [R,page 290]). However, standard regularity theory yields the estimate for general s > 0. For a selfcontained proof, one may refer to [BD2].In the following corollary, parts (i) and (ii) were explicitly stated in [R]; the commutation relationsfollow from the definitions; the other parts follow from the hypoelliptic estimates by standardarguments. Corollary 2.6.6 (Rumin) . Let ( M n +1 , η ) be a compact contact manifold with adapted metric g and the associated complex ( R ∗ , d R ) ; let ∆ R be the associated Laplacian.(i) The cohomology of the complex is finite dimensional and represented by ∆ R -harmonic forms.(ii) There exist operators G R , H R : R k → R k such thatId = G R ∆ R + H R = ∆ R G R + H R , inducing the orthogonal decompositions: R k = ker ∆ R ⊕ range ∆ R = ker ∆ R ⊕ (range d R ⊕ range δ R ) . In particular, each α ∈ R k has a Hodge decomposition α = (cid:40) H R ( α ) ⊕ ( n − k ) G R d R δ R ( α ) ⊕ ( n − k + 1) G R δ R d R ( α ) for k < nH R ( α ) ⊕ G R ( d R δ R ) ( α ) ⊕ G R D ∗ R D R ( α ) for k = n . (iii) The following commutation relations are satisfied:For P any of the operators d R , δ R , D R , or D ∗ R , P H R = H R P = 0 ; for α ∈ R k , d R G R ( α ) = (cid:18) n − k − n − k + 1 (cid:19) G R d R ( α ) for k ≤ n − G R ( d R δ R d R )( α ) for k = n − , and δ R G R ( α ) = (cid:18) n − k + 2 n − k (cid:19) G R δ R ( α ) , for k ≤ n − . (iv) For α ∈ R k , (cid:40) (cid:107) G R α (cid:107) s +2 ≤ c s (cid:107) α (cid:107) s for k (cid:54) = n, ( n + 1) (cid:107) G R α (cid:107) s +4 ≤ c s (cid:107) α (cid:107) s for k = n, ( n + 1) and (cid:107) H R ( α ) (cid:107) s ≤ c s (cid:107) α (cid:107) s .Moreover, since the space of harmonic forms is finite dimensional, (cid:107) α (cid:107) s ≤ c s (cid:107) α (cid:107) for all α ∈ ker ∆ R . In particular, H R ( α ) is of class C ∞ for all α ∈ Γ s ( R k ) . Characterization of contact vector fields. In this section, we present a characterizationof the closed subspace Γ scont ( T M ) ⊂ Γ s ( T M ) of Γ s contact vector fields in terms of the Hodgedecomposition of the Rumin complex. We use this characterization in Section 4 to give a param-eterization of the space of contact diffeomorphisms near the identity by contact vector fields near0. We begin by recalling a few well known facts about contact vector fields. Recall that a smoothvector field X is called a contact vector field if and only if L X ( η ) = 0 mod η (or, equivalently, π H ( L X ( η )) = 0). Write X in the form X = X T + X H where X H ∈ H and T is the Reeb vectorfield, and use the identity L X ( η ) = X dη + d ( X η ) = X H dη + dX to conclude that X is a contact vector field if and only if it satisfies the standard identity − X H dη + T ( X ) η = dX . Recalling that R = H ∗ and d R f = π H ( df ), for f ∈ C ∞ ( M ) = R , we can express this charac-terization in terms of the Rumin complex as follows: X is a contact vector field if and only if itsatisfies the identity(2.7.1) d R X = − X H dη . Next observe that dη defines a vector bundle map( ) (cid:91) : T M → H ∗ : X (cid:55)→ X (cid:91) = X dη whose restriction to H ⊂ T M is an isomorphism between the contact distribution and its dualspace, and let ( ) (cid:93) : H ∗ → H : φ (cid:55)→ φ (cid:93) denote its inverse. The map C ∞ ( M, R ) → C ∞ cont : g (cid:55)→ X g = gT − ( d R g ) (cid:93) is an isomorphism between the space of smooth functions on M and the space of smooth contactvector fields. This map then extends to an isomorphism between the weighted spaces:(2.7.2) Γ s +1 ( M ) → Γ scont ( T M ) : g (cid:55)→ X g = gT − ( d R g ) (cid:93) . ONTACT DIFFEOMORPHISMS 19 The only new result here is the gain of one derivative in the Folland-Stein spaces, which followseasily from the two inclusions g ∈ Γ s and d R g ∈ Γ s .We can now express the condition for X to be a contact vector field in terms of the harmonicdecomposition. Notice, in particular, the additional regularity in X , the Reeb component of X .(The restriction s ≥ n + 4 below ensures that X is of class C .) Lemma 2.7.3. A vector field X ∈ Γ s ( T M ) , s ≥ n + 4 , is contact if and only if it satisfies eachof the following three conditions X = H R ( X ) − ( n + 1) G R δ R ( X dη )(a) H R ( X dη ) = 0(b) (cid:40) d R ( X dη ) = 0 for n > D R ( X dη ) = 0 for n = 1 . (c) Moreover, if X ∈ Γ s ( T M ) is a contact vector field, then X ∈ Γ s +1 ( M ) .Proof. Suppose that X ∈ Γ s ( T M ). Applying the Hodge decomposition to Equation (2.7.1) showsthat X is a contact vector field if and only if it satisfies each of the three conditions H R ( d R X + X dη ) = 0 , δ R ( d R X + X dη ) = 0 , and (cid:40) d R ( d R X + X dη ) = 0 if n > D R ( d R X + X dη ) = 0 if n = 1The middle equation is equivalent to G R δ R ( d R X + X dη ) = 1( n + 1) ( G R ∆ R X ) + G R δ R ( X dη ) = 0 , from which the conditions (2.7.3) follow. Finally, suppose that X ∈ Γ s ( T M ) is a contact vectorfield. By (a) and Corollary 2.6.6, the function X is an element of Γ s +1 ( M ). (cid:3) The topological group of contact diffeomorphisms Let D s ( M ) ⊂ Γ s ( M, M ), s ≥ n + 4 denote the subspace of Γ s -diffeomorphisms of M . It is wellknown that the space of C -diffeomorphisms is an open subset of the space C ( M, M ) of C -maps.Moreover, since s ≥ n + 4, there is a continuous inclusion Γ s ( M, M ) ⊂ C ( M, M ). It follows that D s ( M ) is an open subset of Γ s ( M, M ). It is, therefore, a smooth infinite dimensional manifold; butit is not a group because D s ( M ) is not closed under composition (see Remark 2.1.1).Let D ∞ cont ( M ) ⊂ D s ( M ) denote the subspace of C ∞ contact diffeomorphisms. By definition, thespace of Γ s contact diffeomorphisms of M is the closed subspace D scont ( M ) := D ∞ cont ( M ) ⊂ D s ( M ).We show in this section that D scont ( M ) is closed under composition and inversion, and that bothoperations are continuous. Consequently, D scont ( M ) is a topological group for s ≥ n + 4. InSection 4, we prove that D scont ( M ) is a smooth Hilbert manifold (see Theorem 4.4.1). Our approachin this section and in the following section parallels the treatment of the full diffeomorphism groupgiven by Ebin [E].3.1. Continuity of composition. To prove that composition is continuous, it is sufficient towork locally. Consider open domains D (cid:98) R n +1 and (cid:101) D (cid:98) R n +1 . By the Sobolev lemma (seeRemark 2.4.3), there is a continuous inclusion Γ s ( D, R n +1 ) ⊂ C ( D, R n +1 ). Consequently, thetopological subspace D scont ( D, ˜ D ) of C contact diffeomorphisms f with f ( ¯ D ) ⊂ (cid:101) D is well defined. Proposition 3.1.1. Let s ≥ n + 4 , and let f ∈ D scont ( D, ˜ D ) and g ∈ Γ k ( (cid:101) D, R m ) for k ≤ s .Let D (cid:48) (cid:98) D be an open set. Then the restriction to D (cid:48) of the composition g ◦ f is an element of Γ k ( D (cid:48) , R m ) . Moreover, the map µ : D scont ( D, ˜ D ) × Γ k ( (cid:101) D, R m ) → Γ k ( D (cid:48) , R m ) : ( f, g ) (cid:55)→ g ◦ f is continuous.Proof. Our proof mimics the proof of Ebin [E, Lemma 3.1] in the case of diffeomorphisms of amanifold. It proceeds by induction on k .For k = 0, we first note that (cid:107) g ◦ f (cid:107) D (cid:48) , < ∞ for any D (cid:48) (cid:98) D . Since f is a C diffeomorphismon D , its Jacobian determinant J f is continuous and bounded below on ¯ D (cid:48) by a positive constant;and (by the change of variables formula for integration) (cid:107) g ◦ f (cid:107) D (cid:48) , = (cid:90) D (cid:48) ( g ◦ f ) dV = (cid:90) f ( D (cid:48) ) g (1 /J f ◦ f − ) dV < ∞ . To prove continuity at ( f, g ), choose (cid:15) > 0. We will show that (cid:107) g (cid:48) ◦ f (cid:48) − g ◦ f (cid:107) D (cid:48) , < (cid:15) for ( f (cid:48) , g (cid:48) )sufficiently near ( f, g ). To see this, choose δ > ¯ D (cid:48) ( 1 /J ( f (cid:48) ) , /J ( f )) < (cid:15)/δ whenever (cid:107) f (cid:48) − f (cid:107) D,s < δ . Also choose a smooth function g ∞ on the closure of (cid:101) D such that (cid:107) g − g ∞ (cid:107) (cid:101) D, < δ ,and set M = (cid:32) max x ∈ (cid:101) D n +1 (cid:88) i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂g ∞ ( x ) ∂x i (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) . Then (cid:107) g ◦ f − g (cid:48) ◦ f (cid:48) (cid:107) D (cid:48) , ≤ (cid:107) g ◦ f − g ∞ ◦ f (cid:107) D (cid:48) , + (cid:107) g ∞ ◦ f − g ∞ ◦ f (cid:48) (cid:107) D (cid:48) , + (cid:107) g ∞ ◦ f (cid:48) − g ◦ f (cid:48) (cid:107) D (cid:48) , + (cid:107) g ◦ f (cid:48) − g (cid:48) ◦ f (cid:48) (cid:107) D (cid:48) , ≤ (cid:15)δ (cid:90) (cid:101) D | g − g ∞ | dV + M · (cid:107) f − f (cid:48) (cid:107) D, + (cid:15)δ (cid:90) (cid:101) D | g − g ∞ | dV + (cid:15)δ (cid:90) (cid:101) D | g − g (cid:48) | dV ≤ (cid:15) + M · (cid:107) f − f (cid:48) (cid:107) D, + (cid:16) (cid:15)δ (cid:17) (cid:107) g − g (cid:48) (cid:107) (cid:101) D, . Now let δ (cid:48) = min (cid:16) δ, (cid:15)M (cid:17) . Then the last line is bounded by 4 (cid:15) provided that f (cid:48) and g (cid:48) satisfy theinequalities (cid:107) f − f (cid:48) (cid:107) D, < δ (cid:48) and (cid:107) g − g (cid:48) (cid:107) (cid:101) D, < δ (cid:48) . Assume that for some k ≥ D , D (cid:48) (cid:98) D , and (cid:101) D . We first show that g ◦ f is an element of Γ k +1 ( D (cid:48) ), k + 1 ≤ s , for all g ∈ Γ k +1 ( (cid:101) D, R m ). To do this, we need only showthat X j ( g ◦ f ) is in Γ k ( D (cid:48) ) for 1 ≤ j ≤ n , where X j are the horizontal vector fields defined inSection 1.1. Begin by observing that since f is a contact diffeomorphism, it’s derivative f ∗ respectsthe contact distribution on R n +1 :(3.1.2) f H, ∗ : H x → H f ( x ) : X j ( x ) (cid:55)→ f ∗ ( X j ( x )) = n +1 (cid:88) i =1 A ij ( x ) X i ( f ( x )) , ≤ j ≤ n , ONTACT DIFFEOMORPHISMS 21 where A ij ∈ Γ s − ( D, R ) depend continuously on f . This permits us to compute as follows using thechain rule:(3.1.3) X j ( g ◦ f ) = dg ( f ∗ ( X j )) = n (cid:88) i =1 A ij · X i ( g ) ◦ f . By the induction hypothesis, X i ( g ) ◦ f ∈ Γ k ( D (cid:48)(cid:48) ) for any open set D (cid:48)(cid:48) such that D (cid:48) (cid:98) D (cid:48)(cid:48) (cid:98) D .Since s − ≥ n + 3, we can apply Lemma 2.2.1 to the products A ij · ( X i ( g ) ◦ f ) conclude that A ij · ( X i ( g ) ◦ f ) ∈ Γ k ( D (cid:48) ), which in turn shows that X j ( g ◦ f ) ∈ Γ k ( D (cid:48) ). To complete the inductionstep, we have to prove continuity of composition. First note that if g (cid:48) is near g in Γ k +1 ( (cid:101) D )then X i ( g (cid:48) ) is near X i ( g ) in Γ k ( (cid:101) D ). Now choose a fixed open set D (cid:48)(cid:48) with D (cid:48) (cid:98) D (cid:48)(cid:48) (cid:98) D . By theinduction hypothesis, if f (cid:48) is near f in Γ s ( D ), then X i ( g (cid:48) ) ◦ f (cid:48) is near X i ( g ) ◦ f in Γ k ( D (cid:48)(cid:48) ). But then byLemma 2.2.1 it follows that X j ( g (cid:48) ◦ f (cid:48) ) = n (cid:88) i =1 ( A ij ) (cid:48) · (cid:0) X i ( g (cid:48) ) ◦ f (cid:48) (cid:1) is near X j ( g ◦ f ) = n (cid:88) i =1 A ij · ( X i ( g ) ◦ f )in Γ k ( D (cid:48) ). (cid:3) Corollary 3.1.4. Let M be a compact contact manifold of dimension n +1 , and let N be a smoothmanifold of dimension m . Then the composition map µ : D scont ( M ) × Γ k ( M, N ) → Γ k ( M, N ) : ( F, G ) (cid:55)→ G ◦ F is continuous for n + 4 ≤ k ≤ s . In case N = R m , the map is continuous for n + 4 ≤ s and ≤ k ≤ s .Proof. In case N = R m , choose s ≥ n + 4 and 0 ≤ k ≤ s . Continuity of µ follows easily fromthe previous proposition. We next consider the case where N is an arbitrary smooth manifold andthe definition of Γ k ( M, N ) requires that 2 n + 4 ≤ k . Fix F ∈ D scont ( M ) and G ∈ Γ k ( M, N ). Firstnote that the restriction 2 n + 4 ≤ k ≤ s ensures that the spaces D scont ( M ) and Γ k ( M, N ) are bothwell defined and that both F and G are of class C . Choose adapted atlases { ( φ α , U α , D α ) } and { ( (cid:101) φ α , (cid:101) U α , (cid:101) D α ) } for M , and charts { ψ α , V α , B α } for N , such that for all α , F ( U α ) ⊂ (cid:101) U α , F α ( D α ) ⊂ (cid:101) D α , and G ( (cid:101) U α ) ⊂ V α , where F α = (cid:101) φ α ◦ F ◦ φ − α ∈ Γ s ( U α , (cid:101) U α ) and G α = ψ α ◦ G ◦ (cid:101) φ − α ∈ Γ k ( (cid:101) U α , R m ). Set H α = G α ◦ F α (= ψ α ◦ G ◦ F ◦ φ − α ). By Proposition 3.1.1, H α ∈ Γ k ( (cid:101) U α , R m ) for all α , showing that G ◦ F is anelement of Γ k ( M, N ). To prove continuity of µ , consider the open neighbourhoods of F , G , and H = G ◦ F = µ ( F, G ) O ( F, (cid:15), { U α } ) = { F (cid:48) ∈ D scont ( M ) : F (cid:48) ( U α ) ⊂ (cid:101) U α for all α , max α (cid:107) F (cid:48) α − F α (cid:107) U σ ,s < (cid:15) } ,O ( G, (cid:15), { (cid:101) U α } ) = { G (cid:48) ∈ Γ k ( M, N ) : G (cid:48) ( (cid:101) U α ) ⊂ V α for all α , max α (cid:107) G (cid:48) α − G α (cid:107) (cid:101) U σ ,k < (cid:15) } ,O ( H, (cid:15), { U α } ) = { H (cid:48) ∈ Γ k ( M, N ) : G (cid:48) ( U α ) ⊂ V α for all α , max α (cid:107) H (cid:48) α − H α (cid:107) U α ,k < (cid:15) } . By definition of the topology of Γ k ( M, N ), every open neighbourhood of H contains a set of theform O ( H, (cid:15), { U α } ) for sufficiently small (cid:15) . Moreover, by Proposition 3.1.1, for every (cid:15) > δ > H (cid:48) = G (cid:48) ◦ F (cid:48) ∈ O ( H, (cid:15), { U α } )for all F (cid:48) ∈ O ( F, δ, { U α } ) and G (cid:48) ∈ O ( G, δ, { (cid:101) U α } ). Therefore, µ (cid:16) O ( F, δ, { D α } ) × O ( G, δ, { (cid:101) D α } ) (cid:17) ⊂ O ( G ◦ F, (cid:15), { U α } ) . This completes the proof of continuity of µ . (cid:3) Continuity of inversion. The proof of continuity of inversion relies on the next lemma. Lemma 3.2.1. Let f ∈ D scont ( D, ˜ D ) , s ≥ n + 4 be a contact diffeomorphism, with f ( D ) (cid:98) (cid:101) D ,and let (cid:101) D (cid:48) (cid:98) f ( D ) . Then f − ∈ D scont ( M )( (cid:101) D (cid:48) , D ) . Moreover, the map ι : f (cid:48) (cid:55)→ f (cid:48)− , f (cid:48) ∈ { f (cid:48) ∈D scont ( D, (cid:101) D ) , : (cid:101) D (cid:48) ⊂ f (cid:48) ( D ) } , is continuous at f (cid:48) = f .Proof. Since s ≥ n + 4, f is a C contact diffeomorphism on every compact subset of D . Hence, f − is a C contact diffeomorphism on every open, compactly contained, subset of f ( D ). Let A : D → GL (2 n ) be the matrix valued function defined by A ij , where A ij ∈ Γ s − ( D ) are defined asin the proof of Proposition 3.1.1. Because the process of inverting A only involves multiplication,addition, and division of functions in Γ s − , and because s − > n + 3, we can invoke Lemmas 2.2.1and 2.2.2 to conclude that A − ∈ Γ s − ( D (cid:48) , GL (2 n )) for all D (cid:48) (cid:98) D .Next observe that Equation (3.1.3) with g replaced by g ◦ f − assumes the form X j ( g ) = n (cid:88) i =1 A ij · X i ( g ◦ f − ) ◦ f . Multiplying by B = A − and composing with f − then yields the formula(3.2.2) X k ( g ◦ f − ) = (cid:88) j ( B jk · X j ( g )) ◦ f − . To show that f − ∈ Γ s ( (cid:101) D (cid:48) , R n +1 ) for every open set (cid:101) D (cid:48) (cid:98) f ( D ), it suffices to show that X I ( f − ) ∈ Γ s − k ( (cid:101) D (cid:48) ) for every multi-index I with | I | = k (see Section 1.1). Following the argument on [E,page 17], we proceed by induction on k to show that for all I with | I | = k (3.2.3) X I ( f − ) = g I ◦ f − with g I ∈ Γ s − k .To see that (3.2.3) holds for k = 1, let g = id R n +1 in (3.2.2) to get X i ( f − ) = (cid:88) j ( B ji · X j ( id R n +1 ) ◦ f − := g i ◦ f − , and recall that B ∈ Γ s − to conclude that g i ∈ Γ s − . Now assume that (3.2.3) holds for s > k > I = ( i, J ), for J a multi-index with | J | = k . Then applying (3.2.2) gives(3.2.4) X I ( f − ) = X i ( g J ◦ f − ) = (cid:88) j ( B ji · X j ( g J )) ◦ f − := g I ◦ f − . Since B ji ∈ Γ s − , s − ≥ n + 3 and X j ( g J ) ∈ Γ s − k − , we can invoke Lemma 2.2.1 to conclude that X I ( f − ) is of the form g I ◦ f − , for g I ∈ Γ s − k − . This completes the induction step. We now knowthat X I ( f − ) = g I ◦ f − with g I ∈ Γ s − k ⊂ Γ , for all I with | I | ≤ s . But since f − is of class C ,the composition X I ( f − ) = g I ◦ f − is also in Γ for all I with | I | ≤ s . Hence, f − is in Γ s ( (cid:101) D (cid:48) , D )for all (cid:101) D (cid:48) (cid:98) f ( D ).To prove continuity of the map ι : f (cid:48) (cid:55)→ f (cid:48)− , we first show by finite induction that the map f (cid:48) (cid:55)→ g (cid:48) I := X I ( f (cid:48)− ) ◦ f (cid:48) ∈ Γ ( (cid:101) D (cid:48) , D )depends continuously on f (cid:48) ∈ D scont for all I with | I | ≤ s . Let k = 1, and note that by definitionof A (see Equation (3.1.2)), the assignment f (cid:48) (cid:55)→ A (cid:48) (cid:55)→ B (cid:48) = A (cid:48)− ∈ Γ s − depends continuously on ONTACT DIFFEOMORPHISMS 23 f (cid:48) ∈ D scont . Hence, g (cid:48) i ∈ Γ s − depends continuously on f (cid:48) ∈ D scont . Now assume that f (cid:48) (cid:55)→ g (cid:48) J ∈ Γ s − k depends continuously on f (cid:48) for all J , | J | = k . Set I = ( i, J ). Then X i ( g (cid:48) J ) ∈ Γ s − k − dependscontinuously on f (cid:48) . Hence by (3.2.4), g (cid:48) I ∈ Γ s − k − depends continuously on f (cid:48) , completing theinduction step.Thus, for any multi-index I with | I | ≤ s , (cid:107) X I ( f (cid:48)− − f − ) (cid:107) (cid:101) D (cid:48) , = (cid:107) g (cid:48) I ◦ f (cid:48)− − g I ◦ f − ) (cid:107) (cid:101) D (cid:48) , ≤ (cid:107) g (cid:48) I ◦ f (cid:48)− − g I ◦ f (cid:48)− (cid:107) (cid:101) D (cid:48) , + (cid:107) g I ◦ f (cid:48)− − g I ◦ f − (cid:107) (cid:101) D (cid:48) , . Because the map f (cid:48) (cid:55)→ f (cid:48)− is continuous in the C -topology, by making making (cid:107) f (cid:48) − f (cid:107) (cid:101) D (cid:48) ,s sufficiently small, we ensure that (cid:107) X I ( f (cid:48)− − f − ) (cid:107) (cid:101) D (cid:48) , is arbitrarily small for all I with | I | ≤ s .This concludes the proof of continuity of ι . (cid:3) Theorem 3.2.5. Let s ≥ n + 4 . Then D scont ( M ) is a topological group with group multiplication µ : D scont ( M ) × D scont ( M ) → D scont ( M ) : ( F, G ) (cid:55)→ G ◦ F and group inverse ι : D scont ( M ) → D scont ( M ) : F (cid:55)→ F − . Proof. Continuity of µ is contained in Corollary 3.1.4. Continuity of ι follows from Lemma 3.2.1by an argument similar to the one used in the proof of Corollary 3.1.4. In brief, for fixed F ,choose adapted atlases { ( φ α , U α , D α ) } , and { ( φ (cid:48) α , U (cid:48) α , D (cid:48) α ) } such that D (cid:48) α ⊂ F α ( D α ) for all α . Thenby Lemma 3.2.1, for all (cid:15) > δ > G ∈ O ( F, δ ) we have G − ∈ O ( F − , (cid:15) ). (cid:3) The smooth manifold of contact diffeomorphisms In this section, we obtain a local coordinate chart for the set of contact diffeomorphisms in aneighbourhood of the identity. As a corollary, we show that for s ≥ n + 4, the topological manifold D scont ( M ) is a smooth submanifold of the smooth manifold D s ( M ) of Γ s diffeomorphisms of M .4.1. The smooth manifold of contact diffeomorphisms. We begin by constructing a smoothatlas for D s ( M ). Our construction is based on the following well known parameterization of smoothdiffeomorphisms near the identity diffeomorphism by smooth vector fields. Fix a C ∞ metric adaptedto the contact structure (see Section 1.1), and let exp : T M → M denote its exponential map.Recall that exp is the C ∞ map defined by the formulaexp( X ) := γ x,X (1)where γ x,X : R → M is the unique geodesic curve with γ x,X (0) = x , γ (cid:48) (0) = X ∈ T x M . Also recallthat for | X | sufficiently small, | X | is equal to the Riemannian distance between x and exp( X ).Next consider the map χ from the space of C -vector fields to the space of C -maps χ : C ( T M ) → C ( M, M ) : X (cid:55)→ F X , where F X is the C map defined by composition(4.1.1) F X : M X −→ T M exp −→ M . By compactness of M , there is a number r > r apart are joined by a unique length minimizing geodesic. Let B r M ⊂ T M denote the bundle over M of tangent vectors of length less than r . It is a well known theorem in Riemannian geometry that χ restricts to a diffeomorphism between the space C ( B r M ) of C -vector fields of length lessthan r and the open set { F ∈ C ( M, M ) : dist M ( x, F ( x )) < r for all x ∈ M } , where dist M denotes Riemannian distance. Proposition 4.1.2. For s ≥ n + 4 , the map χ restricts to a smooth map χ s : Γ s ( T M ) → Γ s ( M, M ) : X (cid:55)→ F X = exp ◦ X on the space of Γ s -vector fields. Moreover, there is an open neighbourhood U ⊂ Γ n +4 ( T M ) of thezero section, such that for all s ≥ n + 4 , χ s restricts to a diffeomorphism χ s : U s ≡ U ∩ Γ s ( T M ) → D s ( M ) between U s and a neighbourhood of the identity id M ∈ D s ( M ) .Proof. First observe that a map F : M → M can be viewed as a section of the trivial fibre bundle π M : M × M → M : ( x, y ) (cid:55)→ x and that exp defines a smooth map of fibre bundles (cid:103) exp : T M → M × M : X (cid:55)→ ( π ( X ) , exp( X ))where π : T M → M is projection onto the base point. Smoothness of χ s then follows fromProposition 2.3.5. Let U r M = { ( x, y ) ∈ M × M : dist M ( x, y ) < r } . Then (cid:103) exp : B r M → U r M is asmooth fibre bundle isomorphism. By Proposition 2.4.2, the restriction of χ to Γ s -spaces χ s : Γ s ( B r M ) → Γ s ( U r M )is therefore a diffeomorphism. To complete the proof, let U denote the preimage of the open setΓ n +4 ( U r M ) ∩ D n +4 ( M ) under χ n +4 . (cid:3) Remark 4.1.3. We can use χ s to construct a smooth atlas for smooth manifold D s ( M ). Let G be a smooth diffeomorphism of M . By Corollary 3.1.4, that composition on the left with G givesa smooth diffeomorphism of D s ( M ). Consequently, the map χ sG := L G ◦ χ s : U s → D s ( M ) : X (cid:55)→ G ◦ F X is a local diffeomorphism. Since the set of C ∞ diffeomorphisms of M is dense in D s ( M ), letting G range over all diffeomorphisms gives a smooth atlas. Since composition of smooth maps is smooth,smoothness of the transition functions is automatic. Remark 4.1.4. We recall the standard construction of the tangent bundle π D ( M ) : T D s ( M ) →D s ( M ) (see [Ham, Pal] for background). To get a tangent vector to D s ( M ) at F ∈ D s ( M ), let t (cid:55)→ F t be a smooth curve in D s ( M ) passing through F . Then F t is a smooth family of C diffeomorphisms, so we can differentiate pointwise with respect to t to obtain the vector field X : M → T M : x (cid:55)→ ˙ F ( x ) ∈ T M F ( x ) over F , where we have used the notation ˙ F ( x ) := dF t ( x ) dt (cid:12)(cid:12)(cid:12) t =0 . Conversely, given a vector field X ∈ Γ s ( M, T M ) with π ◦ X = F , observe that for small t the composition F t : M tX −→ T M exp −→ M is a smooth family in D s ( M ) and that X = ˙ F . An easy way to obtain the manifold structure onthe total space T D s ( M ) is to note that by Corollary 2.4.5, composition with the projection map π : T M → M induces a smooth map L sπ : Γ s ( M, T M ) → Γ s ( M, M ). Let T D s ( M ) be the preimageof D s ( M ) under L sπ and let π D ( M ) = L sπ . ONTACT DIFFEOMORPHISMS 25 Characterization of contact diffeomorphisms. Recall that a diffeomorphism F is a con-tact diffeomorphism if and only if the pullback F ∗ η is a multiple of η . Since this condition isequivalent to the equation π H ( F ∗ η ) = 0, where π H : T ∗ M → H ∗ is the quotient map, the space ofΓ s -contact diffeomorphisms near the identity is parameterized by the subspace(4.2.1) V s ≡ { X ∈ U s : π H ( F ∗ X η ) = 0 } ⊂ Γ s ( T M ) . It is convenient to view the equation π H ( F ∗ X η ) = 0 in terms of the Rumin complex. Notice that R ≡ H ∗ , hence F X ∈ U s is a contact diffeomorphism if and only if it satisfies the equation π R ( F ∗ X η ) = 0 . This suggests studying the non-linear differential operator X (cid:55)→ π R ( F ∗ X η )in more detail. Our goal is to show that(4.2.2) π R ( F ∗ X η ) = π R L X η + π R ◦ Q η ( X ) , where L X denotes Lie differentiation with respect to the vector field X and Q η ( X ) is a smoothdifferential operator that vanishes to second order as X → 0. Part (ii) of the next propositionshows that X (cid:55)→ π R ( F ∗ X η ) is a smooth differential operator of contact over 1; that it has the formof Equation (4.2.2) is a corollary to Lemma 4.2.7. Proposition 4.2.3. The following maps are smooth (non-linear) differential operators for all s ≥ n + 4 : (i) Γ s ( T M ) → Γ s − ( T ∗ M ) : X (cid:55)→ F ∗ X η , (ii) Γ s ( T M ) → Γ s − ( R ) : X (cid:55)→ π R F ∗ X η , (iii) Γ s ( T M ) → Γ s − ( R ) : X (cid:55)→ d R ( π R F ∗ X η ) = π R ( dF ∗ X η ) = π R F ∗ X ( dη ) , for n > .Proof. View F X : M → M as a section of the trivial bundle M × M → M . Since s ≥ n + 4,Example 2.5.9 parts (iv) and (v) apply to yield (i) and (ii). To prove (iii), apply Example 2.5.9 (v)to the smooth 2-form dη to conclude that the map X (cid:55)→ π H ( F ∗ X dη )is a smooth differential operator of contact order 1. Next observe that π R = τ ◦ π H , where τ is thequotient map given by (2.6.1). Finally, since d R β = π R ( dβ ), the composition X (cid:55)→ π H ( F ∗ X ( dη )) (cid:55)→ τ ( π H ( F ∗ X dη )) = π R F ∗ X ( dη ) = π R d ( F ∗ X η )is also a smooth differential operator of contact order 1. (cid:3) To show that π H ( F ∗ X η ) has the form of Equation (4.2.2), we work locally, choosing an adaptedatlas φ α : U α → R n +1 for M and a collection of open sets W α (cid:98) U α covering M as in Section 1.1.By compactness of M , there is a constant c > x, X ) ∈ U α for all x ∈ W α , all X ∈ T M x , with | X | < c , and all α .Let X be a C vector field with | X | < c . Fix a chart, say φ α , and set U = U α and W = W α .To simplify notation, we adopt the Einstein summation conventions, letting Roman indices rangefrom 1 to 2 n + 1. Then there exist smooth functions B kij ( x, X ) (locally defined) on T M such that(4.2.4) exp k ( x, X ) = x k + X k + B kij ( x, X ) X i X j . This follows simply from the second order Taylor’s formula with integral remainder for the expo-nential map. Indeed, for fixed X ∈ T x M , let γ ( t ) = exp( x, tX ) be a geodesic. Then(4.2.5) γ k (1) = γ k (0) + ˙ γ k (0) − (cid:90) (1 − t )Γ kij ( γ ( t )) ˙ γ i ( t ) ˙ γ j ( t ) dt , where Γ kij are the Christoffel symbols, and we have used the geodesic equation ¨ γ k + Γ kij ˙ γ i ˙ γ j = 0.Let y = exp( x, X ). Since γ ( t ) = exp( x, tX ) = y ( x, tX ), then˙ γ i ( t ) = ˙ y i ( x, tX ) = ∂y i ( x, tX ) ∂X j X j , and this becomes γ k (1) = γ k (0) + ˙ γ k (0) − (cid:90) (1 − t )Γ kab (exp( x, tX )) ∂y a ∂X i ( x, tX ) ∂y b ∂X j ( x, tX ) X i X j dt , whence(4.2.6) B kij ( x, X ) = − (cid:90) (1 − t )Γ kab (exp( x, tX )) ∂y a ∂X i ( x, tX ) ∂y b ∂X j ( x, tX ) dt . Lemma 4.2.7. Let ψ be a smooth q -form on M and choose a coordinate patch U = U α , with W = W α (cid:98) U . Let c > be chosen so that exp( x, X ) ∈ U for all x ∈ W and all X ∈ T x M with | X | < c . Then there are (locally defined) smooth fibre bundle maps Q ij : BM | W → Λ q M | W and Q ij : BM | W → Λ q − M (cid:12)(cid:12) W , where BM = { X ∈ T M : | X | < c } , such that for any C vector field X : M → BM ⊂ T M theequation F ∗ X ψ = ψ + L X ψ + Q ij ( X ) X i X j + Q ij ( X ) ∧ X i dX j is satisfied on all of W .Proof. Begin with the special case of a 0-form u ∈ C ∞ ( M, R ). Then F ∗ X u ( x ) = u ◦ exp( x, X ), and ap-plying Taylor’s formula with integral remainder to the function f ( t ) = u ( x + tX + t B ij ( x, tX ) X i X j )and setting t = 1 yields the formula( u ◦ exp)( x, X ) = u ( x ) + L X u ( x ) + Q ij ( x, X ) X i X j , for Q ij ( x, X ) smooth functions on BM | W , such that Q ij = Q ji . Next consider the special case ψ = dx k , and compute as follows, using what we have just proved: F ∗ X ( dx k ) = d (cid:16) exp ∗ X x k (cid:17) = d (cid:16) x k + L X ( x k ) + Q ij ( x, X ) X i X j (cid:17) = dx k + L X dx k + d (cid:0) Q ij ( x, X ) X i X j (cid:1) = dx k + L X dx k + d ( Q ij ( x, X )) X i X j + 2 Q ij ( x, X ) X i dX j = dx k + L X dx k + Q ij X i X j + Q ij ( x, X ) X i dX j for Q ij = ∂Q ij ∂x k dx k , Q ij = ∂Q ik ∂X j X k +2 Q ij = ∂ ( Q ik X k ) ∂X j + Q ij . Because every p -form can be expressed asa linear combination of products of terms as above, the general result follows easily by induction. (cid:3) ONTACT DIFFEOMORPHISMS 27 Remark 4.2.8. Henceforth, we will use the notation Q ψ ( X ) := F ∗ X ( ψ ) − ψ − L X ψ to denote the non-linear part of the pull-back F ∗ X ψ . The Lemma states that in local coordinates Q ψ ( X ) = Q ij ( X ) X i X j + Q ij ( X ) ∧ X i dX j , where Q ij and Q ij are smooth functions on BM | W ⊂ T M , which depend on the smooth form ψ .4.3. Parameterization of contact diffeomorphisms. The condition for F X to be a contactdiffeomorphism is the vanishing of the one-form F ∗ X η mod η . Remark 4.2.8 applied to ψ = η andthe identity π R η = 0 show that F X is a contact diffeomorphism if and only if π R L X η + π R Q η ( X ) = 0 . Since Q η ( X ) vanishes to second order at X = 0, the linearization of this equation is π R L X η = 0 , i.e., the condition that X be a contact vector field. This suggests using the implicit functiontheorem in Banach spaces to construct a parameterization of the space of contact diffeomorphismsnear the identity by the space contact vector fields near zero.We are going to construct a smooth map between Hilbert spaces of the formΦ : Γ s ( T M ) → Γ s +1 ( M ) × H s : X (cid:55)→ g X ⊕ γ X , where H s is a second Hilbert space (to be determined), such that(i) F X is a contact diffeomorphism if and only if γ X = 0,(ii) the derivative of Φ is invertible at the origin.By the inverse function theorem, Φ is locally invertible and the map g (cid:55)→ χ s (cid:0) Φ − ( g, (cid:1) gives a smooth parameterization of the contact diffeomorphisms in D scont ( M ) near the identity byreal valued functions in Γ s +1 ( M ) near zero.A natural guess for the map Φ is(4.3.1) X = X T + X H (cid:55)→ ( X , π R F ∗ X η ) , for, as we have already observed, F X is contact if and only if π R F ∗ X η = 0, and X parameterizescontact vector fields (see Section 2.7). Unfortunately, this map is not invertible. Indeed, itslinearization at the origin involves a differential operator that loses too many derivatives.The trick to circumventing this difficulty is to exploit some hidden smoothness in the Hodgedecomposition of one-forms in the Rumin complex. For smooth data, choose Φ to be of the formΦ : (cid:40) C ∞ ( T M ) → C ∞ ( M, R ) ⊕ range( δ R ) ⊕ ker( δ R ) ⊂ C ∞ ( M, R ) ⊕ C ∞ ( M, R ) ⊕ R X (cid:55)→ g X ⊕ α X ⊕ ω X . When n > 1, the Hodge theory shows that the projection π R α ∈ R of a general one-form α ∈ Ω has the decomposition π R α = G R { ( n − d R δ R + n δ R d R } π R α + H R π R α = ( n − G R d R δ R π R α + n G R δ R π R dα + H R π R α. Applying the commutation relations of Corollary 2.6.6(iii), gives π R α = ( n + 1) d R G R δ R π R α + n G R δ R π R dα + H R π R α . This, together with the identity dF ∗ X η = F ∗ X dη give the following decomposition of π R F ∗ X η :(4.3.2) π R F ∗ X η = ( n + 1) d R G R δ R ( π R F ∗ X η ) + nG R δ R ( π R F ∗ X dη ) + H R ( π R F ∗ X η ) . Referring now to our natural guess (4.3.1), we reassemble it using Equation (2.7.3)(a) for X and the identity (4.3.2) to define the map Φ by the formulas(4.3.3) g X = − ( n + 1) G R δ R ( X H dη ) + H R ( X ) ,α X = ( n + 1) G R δ R ( π R F ∗ X η ) ,ω X = n G R δ R d R ( π R F ∗ X η ) + H R ( π R F ∗ X η ) . Indeed, we observe that π R F ∗ X η = d R α X + ω X by Equation (4.3.2), and in the case where X is acontact vector field, then g X = X by Lemma 2.7.3. In the case n = 1, we have to adjust the mapΦ to reflect the Hodge decomposition at R n = R : π R F ∗ X η = G R ( d R δ R ) ( π R F ∗ X η ) + G R D ∗ R D R ( π R F ∗ X η ) + H R ( π R F ∗ X η ) . Applying the commutation relation 2.6.6(iii) to the 0-form δ R ( π R F ∗ X η ), yields the identity π R F ∗ X η = 2 d R G R δ R ( π R F ∗ X η ) + G R D ∗ R D R ( π R F ∗ X η ) + H R ( π R F ∗ X η ) . Because n + 1 = 2, the formulas for g , α , and ω become(4.3.4) g X = − ( n + 1) G R δ R ( X H dη ) + H R ( X ) ,α X = ( n + 1) G R δ R ( π R F ∗ X η ) ,ω X = G R D ∗ R D R ( π R F ∗ X η ) + H R ( π R F ∗ X η ) , and only the formula for ω X has changed. Remark 4.3.5. Observe that by construction π R F ∗ X η = d R α X + ω X . Since d R is injective onrange( δ R ), α X = 0 if and only if d R α X = 0. Consequently, F X is a contact diffeomorphism if andonly if γ X = α X ⊕ ω X = 0 ⊕ Remark 4.3.6. In the case n = 1, the map Φ is, roughly speaking, the same as the map defined in[B]; however, in that paper, the use of the complex Laplacian and complex operators necessitatedan additional splitting into real and imaginary parts—roughly doubling the number of terms. Proposition 4.3.7. Let Φ be the map defined above and let H s := Γ s (range( δ R )) ⊕ Γ s (ker( δ R )) ⊂ Γ s ( M ) ⊕ Γ s ( R ) . Then for s ≥ n + 4 , the map Φ extends to a smooth map Φ : (cid:40) Γ s ( T M ) → Γ s +1 ( M ) ⊕ H s X (cid:55)→ g X ⊕ γ X := g X ⊕ ( α X ⊕ ω X ) The linearization of Φ at the zero vector field is given by d Φ( X ) = (cid:8) − ( n + 1) G R δ R ( X H dη ) + H R ( X ) (cid:9) ⊕ (cid:8) ( n + 1) G R δ R ( d R X + X H dη ) ⊕ ( n G R δ R d R ( X H dη ) + H R ( X H dη )) (cid:9) , for n > (cid:8) − ( n + 1) G R δ R ( X H dη ) + H R ( X ) (cid:9) ⊕ (cid:8) ( n + 1) G R δ R ( d R X + X H dη ) ⊕ ( G R D ∗ R D R ( X H dη ) + H R ( X H dη )) (cid:9) , for n = 1 . ONTACT DIFFEOMORPHISMS 29 Moreover, the linearization of Φ is invertible with inverse given by g ⊕ ( α ⊕ ω ) (cid:55)→ ( g + α ) T + ( − d R g + ω ) (cid:93) where (cid:93) is the isomorphism from horizontal one-forms to horizontal vector fields induced by thetwo-form dη . The proof relies on two lemmas. Lemma 4.3.8. Let F ∈ Γ s ( M, M ) , s ≥ n + 4 , be a diffeomorphism, and let β ∈ Ω k be a smooth k -form, k ≤ n . Then the form π R F ∗ β lies in Γ s − ( R k ) .Proof. This is an immediate corollary to Example 2.5.9 (v) and the observation that for q ≤ n , π R = τ ◦ π H where τ is the quotient map (2.6.1). (cid:3) Lemma 4.3.9. Let n = 1 and let F : M → M be a Γ s (possibly not contact) diffeomorphism, andlet s ≥ n + 4) . Then D R π R F ∗ η is in Γ s − ( R ) .Proof. Recall the definition of the operator D R (see Section 2.6). For any α ∈ R ⊂ Ω ( M ), D R α = d ( (cid:101) α + f η ), where (cid:101) α ∈ Ω ( M ) is any form such that π R ( (cid:101) α ) = α and f ∈ R is the uniquefunction such that η ∧ d ( (cid:101) α + f η ) = 0. Let Λ be the linear isomorphismΛ : R → R = Ω ( M ) : h (cid:55)→ hη ∧ dη . Note that Λ is defined by a smooth vector bundle isomorphism; it, therefore, extends to an isomor-phism between the spaces Γ s ( M ) and Γ s (Λ M ) for all s . Consider now the case α = π R F ∗ η . Thecondition defining f is η ∧ ( F ∗ dη + f dη ) = 0 . By Example 2.5.9(v), η ∧ F ∗ dη is in Γ s − ( R ), forcing f to be in Γ s − ( R ). Compute as follows D R ( π R F ∗ η ) = dF ∗ η + d ( f η ) = F ∗ dη + df ∧ η + f dη . Since wedging with η kills all terms in df ∧ η involving differentiation in directions transverse tothe contact distribution, it follows that d ( f η ) is in Γ s − (Λ M ). Thus D R α is in Γ s − , concludingthe proof of the lemma. (cid:3) Remark 4.3.10. The result of Lemma 4.3.9 is somewhat surprising. Because π R F ∗ η is in Γ s − and D R is an operator of contact order 2, one would expect D R π R F ∗ η only to lie in Γ s − . Proof of Proposition 4.3.7. By Proposition 4.1.2, the map χ s : X (cid:55)→ F X is smooth; and for suffi-ciently small X , F X is a Γ s -diffeomorphism. Recall that π R F ∗ η is in Γ s − ( R ). The linear operators d R , δ R , D R , D ∗ R , H R , and π R are all bounded as maps as follows: d R , δ R : Γ s → Γ s − , H R : Γ s − → Γ s , and π R : Γ s → Γ s ,G R : Γ s ( R k ) → Γ s +2 ( R k ) , for k < n ,D R , D ∗ R : Γ s → Γ s − , G R : Γ s ( R k ) → Γ s +4 ( R k ) for k = n . and the individual terms have the following regularity properties: G R δ R ( π R F ∗ X η ) ∈ Γ s , G R δ R d R ( π R F ∗ X η ) ∈ Γ s , H R ( π R F ∗ X η ) ∈ C ∞ , for n > 1; and for n = 1: G R δ R ( π R F ∗ X η ) ∈ Γ s , G R D ∗ R D R ( π R F ∗ X dη ) ∈ Γ s , H R ( π R F ∗ X η ) ∈ C ∞ . Thus, Φ is smooth and maps between the spaces as indicated. The only nonlinear terms in the map Φ arise from the presence of F ∗ X η . The linearization of thisterm at the zero vector field is L X η = dX + X H dη . When we substitute this into the map Φ,we obtain easily the linearization and its inverse. (cid:3) Let U ⊂ Γ n +4 ( T M ) be an open neighbourhood of the zero section such that F X is a Γ s -diffeomorphism for all X ∈ U s := U ∩ Γ s ( T M ), s ≥ n + 4. The next theorem shows that thesubset V s ⊂ U s on which F X is a contact diffeomorphism is a smooth submanifold which is smoothlyparameterized by the space of Γ s -contact vector fields near zero. Theorem 4.3.11. For U sufficiently small, for all s ≥ n + 4 the set V s = { X ∈ U s : Φ( X ) = ( g X , , } is a smooth submanifold of U s := U ∩ Γ scont ( T M ) , smoothly parameterized by the map Ψ : Γ scont ( T M ) ∩ U s → V s : X (cid:55)→ Φ − ( g X ⊕ . Moreover, the map Ψ is of the form Ψ( X ) = X + B ( X )( X, X ) , where B : (Γ scont ( T M ) ∩ U ) × Γ scont ( T M ) × Γ scont ( T M ) → Γ s ( T M ) is smooth and bilinear in the lasttwo factors.Proof. It suffices to prove the theorem for s = 2 n + 4. That V s ⊂ U s is a smooth submanifoldfollows from Proposition 4.3.7 and the inverse function theorem in Banach spaces. To define Ψ, let π denote the projection(4.3.12) π ( g, γ ) = ( g, , in the notation in the statement of Proposition 4.3.7, and let Ψ = (cid:0) Φ − ◦ π ◦ Φ (cid:1) | Γ scont ( T M ) ∩U . Thesmoothness of Ψ follows from the smoothness of Φ. A simple calculation shows that d Ψ | ( X ) = X , for all X ∈ Γ scont ( T M ), which (together with the inverse function theorem) shows that Ψparameterizes V s . The form of the operator B is given by Taylor’s formula with integral remainderfor smooth operators on Banach spaces: B ( X ) = (cid:90) (1 − t ) D Ψ( tX ) dt . (See e.g. [Ham, Theorem 3.5.6].) (cid:3) Remark 4.3.13. In view of the isomorphism Γ s +1 ( M ) (cid:39) Γ scont ( T M ) given by (2.7.2), the map g (cid:55)→ Ψ( X g ) defines a smooth parameterization of V s by Γ s +1 -functions in a neighbourhood of0 ∈ Γ s +1 ( M, R ).4.4. The smooth structure on the space of contact diffeomorphisms. The map Ψ of Theo-rem 4.3.7 gives a parameterization of the subspace V s ⊂ U s . We now show that this parameteriza-tion in turn induces a smooth structure on the space D scont ( M ) of all Γ s -contact diffeomorphisms. Theorem 4.4.1. Let ( M, η ) be a compact contact manifold. For s ≥ n + 4 , the space of Γ s contactdiffeomorphisms is a smooth Hilbert manifold.Proof. We first show that the intersection of D scont ( M ) with a neighbourhood of the identity is asmooth submanifold of D s ( M ). To see this, let χ s : U s → D s ( M ) be the diffeomorphism onto aneighbourhood of the identity given in Proposition 4.1.2. By Theorem 4.3.11, we can shrink U s if ONTACT DIFFEOMORPHISMS 31 necessary so that V s is a smooth submanifold of U s . Now set O s Id = χ s ( U s ). Since χ s : U s → O s Id is a diffeomorphism and D scont ( M ) ∩ O s Id = χ s ( V s )it follows that D scont ( M ) ∩ O s Id is a smooth submanifold of D s ( M ).Next consider the open set O sF = F ( O s Id ) ⊂ D s ( M ), where F is an arbitrary C ∞ contactdiffeomorphism. Noting that G ∈ D s ( M ) is a contact diffeomorphism if and only if F − ◦ G is acontact diffeomorphism shows that the equality O sF ∩ D scont ( M ) = F ( V s )holds. Finally recall that composition on left with F is a smooth diffeomorphism of D s ( M ) (seeRemark (4.1.3)) to conclude that O sF ∩ D scont ( M ) is a smooth submanifold of D s ( M ).It remains only to show that every element of D scont ( M ) is contained in O sF for some smoothcontact diffeomorphism F . To see this, choose any G ∈ D scont ( M ) and let F k be a sequence ofsmooth contact diffeomorphisms converging to G . Because composition and inversion are contin-uous operations, F − k ◦ G → id M as k → ∞ . Therefore F − k ◦ G ∈ O s Id for k sufficiently large.Consequently G is contained in O sF k = F k ( O sid M ) for k sufficiently large. (cid:3) Remark 4.4.2. We can construct a smooth atlas for D scont ( M ) as follows. By Theorem 4.3.11 andRemark 4.3.13, there is an open neighbourhood O s +1 of 0 ∈ Γ s +1 ( M, R ) such that(4.4.3) χ s +1 cont : O s +1 → D scont ( M ) : g (cid:55)→ χ s ◦ Ψ s ( X g )is a homeomorphism onto O s Id ∩ D scont ( M ). Its inverse is a coordinate chart centered at the identitydiffeomorphism. Composition with a smooth contact diffeomorphism G then yields the map χ s +1 cont,G : O s +1 → D scont ( M ) : g (cid:55)→ G ◦ χ s +1 cont ( g ) , whose inverse is a coordinate chart centered at G . The argument in the last paragraph of the proofof Theorem 4.4.1 shows that the set of all such charts forms a smooth atlas for D scont ( M ).We next address the global topology of T D scont ( M ). Because D scont ( M ) is a closed submanifoldof D s ( M ), there is a smooth inclusion T D scont ( M ) ⊂ T D s ( M ). Using the fact that D s ( M ) is anopen subset of Γ s ( M, M ), and letting Γ s ( M, | T M | ) denote the space of Γ s -maps from M into thetotal space of T M (i.e. forgetting the vector bundle structure on T M ), one sees immediately thatthe tangent bundle of D s ( M ) is the open subset T D s ( M ) = { X ∈ Γ s ( M, | T M | ) : π ◦ X ∈ D s ( M ) } with bundle projection T D s ( M ) → D s ( M ) : X (cid:55)→ π ◦ X . A standard computation with Liederivatives applied to a one-parameter family of contact diffeomorphisms then shows that a Γ s -vector field X : M → | T M | is in T D scont ( M ) if and only if F = π ◦ X is D scont ( M ) and X ◦ F − ∈ Γ s ( T M ) is a contact vector field. As the next proposition shows, T D scont ( M ) is a trivial vectorbundle: Proposition 4.4.4. Let X g denote the contact vector field associated to the generating function g .The map D scont ( M ) × Γ s ( M, R ) −→ T D scont ( M ) : ( g, F ) (cid:55)→ X g ◦ F , is a continuous vector bundle isomorphism. Remark 4.4.5. Composition with a Γ s -contact diffeomorphism is a continuous, but not differen-tiable, operation. Consequently, the trivialization in Proposition 4.4.4 is not smooth. We discussthe smoothness of composition in Section 4.5. Proposition 4.4.4 is a corollary to a more general construction. Let π : E → M be a smoothvector bundle over M , and let | E | denote the total space of E , viewed as a smooth manifold,forgetting its vector bundle structure. Recall that Γ s ( M, | E | ) denotes the space of Γ s -maps from M into | E | . Because π is smooth, Corollary 2.4.5 applies to show that the map L sπ : Γ s ( M, | E | ) → Γ s ( M, M ) : G (cid:55)→ π ◦ G . is smooth. Let Γ s D cont ( M, E ) = ( L sπ ) − ( D scont ( M )) ⊂ Γ s ( M, | E | ) and let(4.4.6) π D : Γ s D cont ( M, E ) −→ D scont ( M )be the restriction of L sπ to Γ s D cont ( M, E ). Notice that the vector bundle structure on E inducesa vector space structure on the the fibres of π D , and we call π D : Γ s D cont ( M, E ) → D scont ( M ) the(vector) bundle of Γ s -sections of E over contact diffeomorphisms. Lemma 4.4.7. Let π : E → M be a smooth vector bundle over M . Then for s ≥ n + 4 , themap Φ E : D scont ( M ) × Γ s ( E ) → Γ s D cont ( M, E ) : ( F, σ ) (cid:55)→ σ ◦ F is a continuous, vector bundleisomorphism between Γ s D cont ( M, E ) and the trivial vector bundle.Proof. Consider first the special case where E → M is the trivial bundle M × R r → M . Thediffeomorphism Γ s ( M, | M × R r | ) (cid:39) Γ s ( M, M ) × Γ s ( M, R r ) restricts to a diffeomorphismΓ s D cont ( M, M × R r ) (cid:39) D scont ( M ) × Γ s ( M, R r )with respect to which Φ M × R r assumes the formΦ M × R r : D scont ( M ) × Γ s ( M, R r ) → Γ s D cont ( M, M × R r ) : ( F, σ ) (cid:55)→ ( F, σ ◦ F ) , with inverseΦ − M × R r : Γ s D cont ( M, M × R r ) → D scont ( M ) × Γ s ( M, R r ) : ( F, σ ) (cid:55)→ ( F, σ ◦ F − ) . Continuity of Φ − M × R r follows from continuity of composition with F (see Corollary 3.1.4); andcontinuity of Φ − M × R r follows from continuity of inversion (see Theorem 3.2.5).Now consider the general case. By construction, Φ E is bijective, preserves basepoint, and is linearon each fibre. To see that Φ E is continuous, note that since the map E → M × E : e (cid:55)→ ( π ( e ) , e )is smooth, so is the induced map ι : Γ s ( E ) (cid:44) → Γ s ( M × E )defined by the formula ι ( σ ) : x (cid:55)→ ( x, σ ( x )). This observation, together with Corollary 2.4.5 impliescontinuity of Φ E . It remains only to show that Φ − E is continuous. Let j : E → M × R r be a smoothvector bundle inclusion into a trivial bundle, and let s : M × R r → E be a smooth vector bundlemap with s ◦ j = id E . Continuity of Φ − E is proved by expressing Φ − E as the following compositionof continuous mapsΓ s D cont ( M, E ) L sj −→ Γ s D cont ( M, M × R r ) (cid:39) D scont ( M ) × Γ s ( M, R r ) Φ − M × R r −→ D scont ( M ) × Γ s ( M, R r ) id D scont ( M ) × L ss −→ D scont ( M ) × Γ s ( M, E ) , concluding the proof of the lemma. (cid:3) We close this section with a formula for the derivative of χ scont , which we need in Section 4.5. For g ∈ O s +1 , we denote by T F g D scont ( M ) the tangent space to D scont ( M ) at the contact diffeomorphism F g , and for h ∈ Γ s +1 ( M, R ), we set Y ( g,h ) = Dχ s +1 contg ( h ) : M → T M , ONTACT DIFFEOMORPHISMS 33 where Y ( g,h ) ∈ T F g D scont ( M ); and we set X ( g,h ) = Y ( g,h ) ◦ F − g ∈ Γ scont ( T M ). Lemma 4.4.8. For s ≥ n + 4 , the map O s +1 × Γ s +2 ( M, R ) → Γ scont ( T M ) : ( g, h ) (cid:55)→ X ( g,h ) is continuous. Moreover, for every g ∈ O s +1 and (cid:15) > , there is a δ > such that (cid:107) X ( g ,h ) − X ( g,h ) (cid:107) s < (cid:15) (cid:107) h (cid:107) s +1 for all g ∈ O s +1 such that (cid:107) g − g (cid:107) s +1 < δ and all h ∈ Γ s +2 ( M, R ) .Proof. Continuity of the map is clear. The estimate is the restatement of the fact that the derivative Dχ scont,g depends continuously on g . (cid:3) Differentiability of composition. We showed in Section 3 that composition µ : D scont ( M ) × Γ k ( M, R ) → Γ k ( M, R ) : ( F, u ) (cid:55)→ u ◦ F is a continuous operation for 2 n + 4 ≤ k ≤ s , but composition is not C , as the following coun-terexample shows. Choose u ∈ Γ k ( M, R ) with T ( u ) / ∈ Γ k ( M R ), where T is the Reeb vector fieldon M . Then the one-parameter family F t of contact diffeomorphisms given by the flow of the Reebvector field T is a smooth curve in D scont ( M ); and differentiability of µ would imply that the limitlim t → µ ( u, F t ) − µ ( u, F ) t = T ( u )would be an element of Γ k ( M, R ). But this contradicts our choice of u . The next theorem showsthat we can recover smoothness by strengthening the regularity assumption on u . Theorem 4.5.1. Let M be a compact contact manifold of dimension n + 1 and let N be a smoothmanifold. Then the map µ : D scont ( M ) × Γ k +2 ( M, N ) → Γ k ( M, N ) : ( F, G ) (cid:55)→ G ◦ F is continuously differentiable for n + 4 ≤ k ≤ s . In case N = R m , µ is continuously differentiablefor ≤ k ≤ s , n + 4 ≤ s .Proof. Assume that the theorem holds in the special case where N = R m , with m arbitrary. Let ι : N (cid:44) → R m be a closed embedding, and let U ⊂ R m be a tubular neighbourhood of N , withprojection map π : U → N . By Corollary 2.4.5, the maps ι and π induce smooth maps (cid:101) ι : Γ k ( M, N ) → Γ k ( M, U ) and (cid:101) π : Γ k ( M, U ) → Γ k ( M, N )for all k ≥ n + 4. Because Γ k ( M, U ) is an open subset of Γ k ( M, R m ), by assumption, we knowthat µ : D scont ( M ) × Γ k +2 ( M, U ) → Γ k ( M, U ) : ( F, u ) (cid:55)→ u ◦ F is a C map. It follows that the composition D scont ( M ) × Γ k +2 ( M, N ) id × (cid:101) ι −→ D scont ( M ) × Γ k +2 ( M, U ) µ −→ Γ k ( M, U ) (cid:101) π −→ Γ k ( M, N )is a C map.It remains to prove the theorem in the case N = R m . Because Γ k ( M, R m ) is the m -th foldproduct of Γ k ( M, R ), we need only prove it for m = 1; and by Remark 4.4.2, it suffices to restrictto an open neighbourhood of the identity in D scont ( M ). Now for O s +1 ⊂ Γ s +1 ( M ) a sufficientlysmall neighbourhood of 0, the map χ s +1 cont : O s +1 −→ D scont ( M ) : g (cid:55)→ F g is a smooth parameterization of a neighbourhood of the identity contact diffeomorphism. (Hereand in the following we set F g = F Ψ( X g ) , where X g denotes the contact vector field with generatingfunction g .) With this notation, the proof reduces to proving that the map µ : Γ k +2 ( M, R ) × O s +1 −→ Γ k ( M, R ) : ( u, g ) (cid:55)→ u ◦ F g is C . The next proposition completes the proof. (cid:3) Proposition 4.5.2. For s ≥ n + 4 and s ≥ k ≥ and for O s +1 ⊂ Γ s +1 ( M, R ) a sufficiently smallneighbourhood of , the map µ : Γ k +2 ( M, R ) × O s +1 −→ Γ k ( M, R ) : ( u, g ) (cid:55)→ u ◦ F g . is C with derivative at ( u, g ) given by the formula Dµ ( u,g ) : ( v, h ) (cid:55)→ v ◦ F g + ( X ( g,h ) du ) ◦ F g , for ( v, h ) ∈ Γ k +2 ( M, R ) × Γ s +1 ( M, R ) .Proof. Because µ is a map between Banach spaces, to show that it is C , we need only show thatthe two partial derivatives of µ with respect to the first and second variables D µ : Γ k +2 ( M, R ) × O s +1 → L (cid:16) Γ k +2 ( M, R ) , Γ k ( M, R ) (cid:17) D µ : Γ k +2 ( M, R ) × O s +1 → L (cid:16) Γ s +1 ( M, R ) , Γ k ( M, R ) (cid:17) exist and are continuous . We shall obtain formulas for D µ and D µ . The formula for Dµ ( u,g ) ( v, h )then follows immediately from the well-known identity Dµ ( u,g ) ( v, h ) = D µ ( u,g ) ( v ) + D µ ( u,g ) ( h ) . To see that D µ exists, notice that by Corollary 3.1.4, µ is continuous. The map µ is linearin the first variable and, therefore, differentiable with respect to the first variable, with derivativegiven by D µ ( u,g ) ( v ) = v ◦ F g . Continuity of D µ is proved in Lemma 4.5.4 below.We next claim that D µ ( u,g ) ( h ) = ( X ( g,h ) du ) ◦ F g . To prove the claim, first notice that because s ≥ n + 4, the functions g , h , u , as well as the map F g are of class at least C . Moreover, because χ s +1 cont is smooth, the family t (cid:55)→ F g + th of contactdiffeomorphisms is a smooth family. Consequently, we can compute pointwise at x ∈ M , employingthe chain rule as follows:(4.5.3) D µ ( u,g ) ( h )( x ) = lim t → u ( F g + th ( x )) − u ( F g ( x )) t = du ( F g + th ( x )) dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = du (cid:16) Dχ s +1 cont,g ( h )( x ) (cid:17) = Y ( g,h ) ( x ) du F g ( x ) = ( X ( g,h ) du ) ◦ F g ( x ) . To prove that µ is differentiable with respect to the second variable we need to verify the formulalim h → (cid:107) u ◦ F g + h − u ◦ F g − ( X ( g,h ) du ) ◦ F g (cid:107) k (cid:107) h (cid:107) s +1 = 0and we need to prove continuity of D µ . We do this in Lemma 4.5.10. (cid:3) We use the notation L ( H , H ) to denote the Banach space of bounded linear maps between Hilbert spaces H and H . ONTACT DIFFEOMORPHISMS 35 Lemma 4.5.4. D µ : Γ k +2 ( M, R ) ×O s +1 → L (cid:0) Γ k +2 ( M, R ) , Γ k ( M, R ) (cid:1) is continuous for s ≥ n +4 and all k ≥ .Proof. By definition of continuity, we must show that for any g ∈ O s +1 and any (cid:15) > 0, there is a δ > (cid:107) v ◦ F g − v ◦ F g (cid:107) k < (cid:15) (cid:107) v (cid:107) k +2 is satisfied for all v ∈ Γ k ( M, R ) and all g ∈ O s +1 with (cid:107) g − g (cid:107) s +1 < δ . We need only prove theestimate for v a smooth test function.Set h = g − g . Because the map g (cid:55)→ F g is smooth, and v is smooth, for fixed x ∈ M , thefunction γ x : t (cid:55)→ v ( F g + th ( x )) is a C function of t . Consequently, we can compute as follows usingthe chain rule: v ◦ F g ( x ) − v ◦ F g ( x ) = (cid:90) ddt v ( F g + th ( x )) dt = (cid:90) ( X ( g + th,h ) dv ) ◦ F g + th ( x ) dt . Viewing t (cid:55)→ ( X ( g + th,h ) dv ) ◦ F g + th as a continuous curve in the Hilbert space Γ k ( M, R ) yieldsthe inequality (cid:107) v ◦ F g ( x ) − v ◦ F g ( x ) (cid:107) k ≤ (cid:90) (cid:13)(cid:13) ( X ( g + th,h ) dv ) ◦ F g + th (cid:13)(cid:13) k dt . Hence, to prove the lemma it suffices to show that, for δ > (cid:107) ( X g,h dv ) ◦ F g (cid:107) k < (cid:15) (cid:107) v (cid:107) k +2 holds for all g ∈ O s +1 and h ∈ Γ s +1 ( M, R ) with (cid:107) g − g (cid:107) s +1 < δ and (cid:107) h (cid:107) s +1 < δ . To obtain (4.5.5)first observe that since interior evaluationΓ s ( T M ) × Γ k ( T ∗ M ) → Γ k ( M, R ) : ( X, β ) (cid:55)→ X β is a smooth bilinear map (see Proposition 2.4.2), the estimate(4.5.6) (cid:107) X β (cid:107) k ≺ (cid:107) X (cid:107) s (cid:107) β (cid:107) k holds. Note also that by continuity of composition (see Corollary 3.1.4) and linearity in v ,(4.5.7) (cid:107) v ◦ F g (cid:107) k ≺ (cid:107) v (cid:107) k for all v . Continuity also shows that we can choose δ > (cid:107) v ◦ F g − v ◦ F g (cid:107) k < (cid:107) v (cid:107) k < δ and (cid:107) g − g (cid:107) s +1 < δ . By linearity in v , setting C = 1 /δ , we get theestimate(4.5.8) (cid:107) v ◦ F g − v ◦ F g (cid:107) k < C (cid:107) v (cid:107) k , provided (cid:107) g − g (cid:107) s +1 < δ . Finally note that because χ s +1 cont is smooth, its derivative Dχ s +1 cont iscontinuous in the operator norm. We can therefore choose δ > (cid:107) X ( g,h ) − X ( g ,h ) (cid:107) s < (cid:15) (cid:107) h (cid:107) s +1 for all h , provided (cid:107) g − g (cid:107) s +1 < δ . Choosing δ > g so that (cid:107) g − g (cid:107) s +1 < δ , we can then estimate as follows: (cid:107) ( X ( g,h ) dv ) ◦ F g (cid:107) k ≤ (cid:107) ( X ( g,h ) dv ) ◦ F g (cid:107) k + (cid:107) ( X ( g,h ) dv ) ◦ F g − ( X ( g,h ) dv ) ◦ F g (cid:107) k ≺ (cid:107) ( X ( g,h ) dv ) ◦ F g (cid:107) k + (cid:107) X ( g,h ) dv (cid:107) k ≺ (cid:107) X ( g,h ) dv (cid:107) k + (cid:107) X ( g,h ) dv (cid:107) k ≺ (cid:107) X ( g,h ) dv (cid:107) k ≺ (cid:107) X ( g,h ) (cid:107) s · (cid:107) v (cid:107) k +2 ≺ ( (cid:107) ( X ( g ,h ) dv ) (cid:107) s + (cid:15) (cid:107) h (cid:107) s +1 ) · (cid:107) v (cid:107) k +2 ≺ (cid:107) h (cid:107) s +1 · (cid:107) v (cid:107) k +26 J. BLAND AND T. DUCHAMP The estimate (4.5.5) follows by decreasing δ , if necessary, and requiring (cid:107) h (cid:107) s +1 < δ . (cid:3) Lemma 4.5.10. For n + 4 ≤ s , k ≤ s , and O s +1 as in Lemma 4.5.4, the derivative D µ :Γ k +2 ( M, R ) × O s +1 → L (cid:0) Γ s +1 ( M, R ) , Γ k ( M, R ) (cid:1) exists, is continuous, and given by the formula D µ ( u,g ) ( h ) = Y ( g,h ) ( du ◦ F g ) , for g ∈ O s +1 , u ∈ Γ k ( M, R ) , and h ∈ Γ s +1 ( M, R ) .Proof. To prove that D µ exists, choose ( u, g ) ∈ Γ k +2 ( M, R ) × O s +1 . We need to show that(4.5.11) lim h → (cid:107) u ◦ F g + h − u ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:107) k (cid:107) h (cid:107) s +1 = 0 . Choose (cid:15) > 0. We need to find δ > (cid:107) u ◦ F g + h − u ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:107) k < (cid:15) (cid:107) h (cid:107) s +1 for (cid:107) h (cid:107) s +1 < δ . To this end, compute as follows for g ∈ O s +1 near g , setting h = g − g : (cid:107) u ◦ F g + h − u ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:107) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) (cid:26) d ( u ◦ F g + th ) dt − ( X ( g ,h ) du ) ◦ F g (cid:27) dt (cid:13)(cid:13)(cid:13)(cid:13) k ≤ (cid:90) (cid:13)(cid:13)(cid:13)(cid:13)(cid:26) d ( u ◦ F g + th ) dt − ( X ( g ,h ) du ) ◦ F g (cid:27)(cid:13)(cid:13)(cid:13)(cid:13) k dt = (cid:90) (cid:13)(cid:13) ( X ( g + th,h ) du ) ◦ F g + th − ( X ( g ,h ) du ) ◦ F g (cid:13)(cid:13) k dt Consequently, to prove (4.5.12), it suffices to find δ > (cid:13)(cid:13) ( X ( g,h ) du ) ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:13)(cid:13) k < (cid:15) (cid:107) h (cid:107) s +1 holds for all (cid:107) g − g (cid:107) s +1 < δ . But by Proposition 4.5.2, we can choose δ > (cid:107) v ◦ F g − v ◦ F g (cid:107) k < (cid:15) (cid:107) v (cid:107) k +2 for all v ∈ Γ k ( M, R ); and with this choice of δ , we can estimate asfollows: (cid:13)(cid:13) ( X ( g,h ) du ) ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:13)(cid:13) k ≤ (cid:13)(cid:13) ( X ( g,h ) du ) ◦ F g − ( X g,h du ) ◦ F g (cid:13)(cid:13) k + (cid:13)(cid:13) ( X ( g,h ) du ) ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:13)(cid:13) k ≺ (cid:15) (cid:107) X ( g,h ) du (cid:107) k + (cid:107) (cid:0) X ( g,h ) − X ( g ,h ) (cid:1) du (cid:107) k ≺ (cid:15) (cid:107) X ( g,h ) du (cid:107) k + (cid:107) X ( g,h ) − X ( g ,h ) (cid:107) s · (cid:107) du (cid:107) k . Finally, using smoothness of χ s +1 cont as we did in Equation (4.5.9) above we can bound the last termas follows, for δ sufficiently small: ≺ (cid:15) (cid:107) X ( g,h ) du (cid:107) k + ( (cid:15) (cid:107) h (cid:107) s +1 ) (cid:107) du (cid:107) k ≺ (cid:15) (cid:107) h (cid:107) s +1 . This concludes the proof of (4.5.12). Continuity of D µ follows from the following estimate: (cid:107) D µ ( g,u ) ( h ) − D µ ( g ,u ) ( h ) (cid:107) k = (cid:107) ( X ( g,h ) du ) ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:107) k ≺ (cid:107) ( X ( g,h ) du ) ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:107) k + (cid:107) ( X ( g ,h ) du ) ◦ F g − ( X ( g ,h ) du ) ◦ F g (cid:107) k ≺ (cid:15) (cid:107) du (cid:107) k (cid:107) h (cid:107) s +1 + C (cid:107) d ( u − u ) (cid:107)(cid:107) h (cid:107) s +1 ≺ (cid:15) (cid:107) h (cid:107) s +1 , which holds for all ( g, u ) with (cid:107) g − g (cid:107) s +1 < δ , (cid:107) u − u (cid:107) k +1 < δ . (cid:3) ONTACT DIFFEOMORPHISMS 37 Some a priori estimates. Theorems 4.3.11 and 4.4.1 state that the nonlinear space of Γ s contact diffeomorphisms is a Hilbert manifold modelled on the linear space of Γ s contact vectorfields. In typical applications, one would like to study the action of the space of contact diffeo-morphisms on some set of structures by comparing it with the linearized action of contact vectorfields. For this strategy to work, it is necessary to show that the error incurred in the linearizationis quadratically small in an appropriate sense. This is the content of the next proposition, whichgives a priori estimates for the quadratic error Ψ( X ) − X = B ( X )( X, X ). We require this result in[BD3] to obtain normal forms for CR structures on compact three dimensional contact manifolds. Proposition 4.6.1. For X ∈ Γ scont ( T M ) ∩ U s , (i) (cid:107) Ψ( X ) − X (cid:107) s ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − . Moreover, for all X , X ∈ Γ scont ( T M ) ∩ U s , (cid:107) (Ψ( X ) − X ) − (Ψ( X ) − X ) (cid:107) s ≺ (cid:107) X − X (cid:107) s − ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s )(ii) + (cid:107) X − X (cid:107) s ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) . Our proof relies on the next two lemmas. The first is a corollary to Lemma 4.2.7 and compactnessof M . (See Remark 4.2.8 for the definition of Q .) Lemma 4.6.2. For c > sufficiently small, the following estimates hold for ψ a fixed smooth q form. For all X ∈ Γ s +2 cont ( T M ) , s ≥ n + 4 , such that | X | < c : (cid:107) F ∗ X ψ (cid:107) s ≺ (cid:107) ψ (cid:107) s + (cid:107)L X ψ (cid:107) s + (cid:107) X (cid:107) s (cid:107) X (cid:107) s +2 , (i) (cid:107) ( F ∗ X ψ ) ∧ η (cid:107) s ≺ (cid:107) ψ ∧ η (cid:107) s + (cid:107)L X ψ ∧ η (cid:107) s + (cid:107) X (cid:107) s (cid:107) X (cid:107) s +1 , (ii) and, for q ≤ n , (cid:107) π R ( F ∗ X ψ ) (cid:107) s ≺ (cid:107) π R ψ (cid:107) s + (cid:107) π R ( L X ψ ) (cid:107) s + (cid:107) X (cid:107) s (cid:107) X (cid:107) s +1 . (iii) Moreover, for q ≤ n , (cid:107) π R ( Q ψ ( X ) − Q ψ ( X )) (cid:107) s ≺ (cid:107) X − X (cid:107) s ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 )+ (cid:107) X − X (cid:107) s +1 ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) , (iv) holds for any two vector fields X i , i = 1 , such that | X i | < c .Proof. Choose an adapted atlas and a constant c > Q ψ . To prove (iii), note that by Lemma 4.2.7, π R ◦ Q ψ is a smooth differential operator of contact order 1; the inequality then follows. Theinequality (iv) follows from the smooth dependence of Q ψ ( X ) on X . (cid:3) Lemma 4.6.3. Choose c > as in the previous lemma. The following estimates are satisfied forany Γ s +2 vector fields X , X i , i = 1 , with | X | < c , | X i | < c , s ≥ n + 4 . If n = 1 then (cid:107) D R π R ( F ∗ X η − η − L X η ) (cid:107) s − = (cid:107) D R π R ( Q η ( X )) (cid:107) s − ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − and (cid:107) D R π R ( Q η ( X ) − Q η ( X )) (cid:107) s − ≺ ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) · (cid:107) X − X (cid:107) s − + ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) · (cid:107) X − X (cid:107) s . If n > then (cid:107) d R π R ( F ∗ X η − η − L X η ) (cid:107) s − = (cid:107) d R π R ( Q η ( X )) (cid:107) s − ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − 18 J. BLAND AND T. DUCHAMP and (cid:107) d R π R ( Q η ( X ) − Q η ( X )) (cid:107) s − ,loc ≺ ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) · (cid:107) X − X (cid:107) s − + ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) · (cid:107) X − X (cid:107) s . Proof. Let φ = F ∗ X η − η − L X η ∈ Γ s (Λ ( M ). By compactness of M , it suffices to obtain localestimates on a coordinate patch W (cid:98) U chosen as in Lemma 4.2.7. In the notation of Lemma 4.2.7,the one form φ can be written φ = Q ij ( X ) X i X j + Q ij ( X ) X i dX j = Q ij,k ( x, X ( x )) X i ( x ) X j ( x ) dx k + Q ij ( x, X ( x )) X i ( x ) dX j ( x ) , where Q ij,k and Q ij are smooth functions on BM | W .Suppose that n = 1. Recall that D R ( π R φ ) is defined as D R π R φ := d ( φ + f η ) = dφ + d R f ∧ η + f ∧ dη . where f ∈ Γ s − ( M ) is the unique function with η ∧ ( dφ + f dη ) = 0. Because the map h (cid:55)→ hη ∧ dη is a smooth linear isomorphism, (cid:107) f (cid:107) s − ≺ (cid:107) f η ∧ dη (cid:107) s − . We can, therefore, estimate as follows: (cid:107) D R π R φ (cid:107) s − ≺ (cid:107) dφ (cid:107) s − + (cid:107) f (cid:107) s − ≺ (cid:107) dφ (cid:107) s − + (cid:107) dφ ∧ η (cid:107) s − . Thus, we need only estimate (cid:107) dφ (cid:107) s − and (cid:107) dφ ∧ η (cid:107) s − : (cid:107) dφ (cid:107) W,s − = (cid:107) d ( Q ij,k ( x, X ) X i X j dx k + Q ij ( x, X ) X i dX j ) (cid:107) W,s − ≺ (cid:107) X (cid:107) W,s − (cid:107) X (cid:107) W,s (cid:107) ( dφ ∧ η ) (cid:107) W,s − ≺ (cid:107) d ( Q ij,k ( x, X ) X i X j dx k + Q ij ( x, X ) X i dX j ) ∧ η (cid:107) W,s − ≺ (cid:107) X (cid:107) W,s − (cid:107) X (cid:107) W,s . The proof of the second inequality follows by similar reasoning.Now suppose that n > 1. Then (cid:107) d R π R ( F ∗ X η − η − L X ) (cid:107) W,s − = (cid:107) π R dφ (cid:107) W,s − = (cid:107) d R ( Q ij,k ( x, X ) X i X j dx k ) + d R ( Q ij ( x, X ) X i ) ∧ d R X j ) (cid:107) W,s − ≺ (cid:107) X (cid:107) W,s (cid:107) X (cid:107) W,s − The proof of the last inequality in the statement of the lemma follows by similar reasoning. (cid:3) Proof of Proposition 4.6.1. Recall the definition of Ψ,(4.6.4) Ψ( X ) − X = Φ − ◦ π ◦ Φ( X ) − X = Φ − ◦ π ◦ Φ( X ) − Φ − ◦ Φ( X )where Ψ is the map defined in Theorem 4.3.7 and where π ( g ⊕ α ⊕ ω ) = g . Because Φ − is smooth(and thus of class C ), Equation (4.6.4) implies the inequality (cid:107) Ψ( X ) − X (cid:107) s ≺ ||| π ◦ Φ( X ) − Φ( X ) ||| s = (cid:107) π ⊥ Φ( X ) (cid:107) s , where π ⊥ ( g ⊕ α ⊕ ω ) := α ⊕ ω and ||| g ⊕ α ⊕ ω ||| s := (cid:107) g (cid:107) s +1 + (cid:107) α ⊕ ω (cid:107) s . Consequently, to provethe estimate (i), we need only estimate the terms in the expansion(4.6.5) π ⊥ Φ( X ) = (cid:40) ( n + 1) G R δ R ( π R F ∗ X η ) ⊕ ( G R D ∗ R D R ( π R F ∗ X η ) ⊕ H R ( π R F ∗ X η )) , for n = 1( n + 1) G R δ R ( π R F ∗ X η ) ⊕ ( nG R δ R d R ( π R F ∗ X η ) ⊕ H R ( π R F ∗ X η )) , for n > ONTACT DIFFEOMORPHISMS 39 given in Theorem 4.3.7. By Lemma 4.2.7 we have the local formula F ∗ X η = η + L X η + Q ij ( x, X ) X i X j + Q ij ( x, X ) X i dX j . Therefore, because X is contact and so π R ( L X η ) = 0, Lemma 4.6.2 implies the estimate (cid:107) π R ( F ∗ X η ) (cid:107) s − ≺ (cid:107) π R η (cid:107) s − + (cid:107) π R ( L X η ) (cid:107) s − + (cid:107) X (cid:107) s (cid:107) X (cid:107) s − = (cid:107) X (cid:107) s (cid:107) X (cid:107) s − . Taking into account the orders of the various operators in Equation (4.6.5) and recalling Lemma 4.6.3,we can estimate as follows:For n ≥ (cid:107) G R δ R π R ( F ∗ X η ) (cid:107) s ≺ (cid:107) π R ( F ∗ X η ) (cid:107) s − ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − (cid:107) H R π R ( F ∗ X η ) (cid:107) s ≺ (cid:107) π R ( F ∗ X η ) (cid:107) s − ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − ;for n = 1, (cid:107) G R D ∗ R D R π R ( F ∗ X η ) (cid:107) s ≺ (cid:107) D R π R ( F ∗ X η ) (cid:107) s − ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − ;and for n > (cid:107) G R δ R d R π R ( F ∗ X η ) (cid:107) s ≺ (cid:107) d R π R ( F ∗ X η ) (cid:107) s − ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − . This concludes the proof of (i).We now show that (cid:107) (Ψ( X ) − X ) − (Ψ( X ) − X ) (cid:107) s ≺ (cid:107) X − X (cid:107) s − ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s )+ (cid:107) X − X (cid:107) s ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) . Set Y i := Ψ( X i ) − X i , i = 1 , 2. Differentiability of the map Φ − implies the inequality (cid:107) Y − Y (cid:107) s = (cid:107) (Ψ( X ) − X ) − (Ψ( X ) − X ) (cid:107) s ≺ (cid:107)| Φ( Y − Y ) (cid:107)| s ;which, by adding and subtracting a term and applying the triangle inequality, implies the inequality(4.6.6) (cid:107) Y − Y (cid:107) s ≺ ||| ( π ⊥ ◦ Φ( X ) − π ⊥ ◦ Φ( X )) ||| s + ||| Φ( Y − Y ) + ( π ⊥ ◦ Φ( X ) − π ⊥ ◦ Φ( X )) ||| s . To conclude the proof, we need only estimate each term on the right-hand side of (4.6.6).To estimate the first term, note that by Lemma 4.2.7 applied to the contact vector fields X i , wehave the local formula π R F ∗ X i η = π R ( η + L X i η + Q η ( X i )) = π R ( Q η ( X i )) . This and Equation (4.6.5) imply the inequalities (cid:107)| π ⊥ ◦ Φ( X ) − π ⊥ ◦ Φ( X ) (cid:107)| s ≺ (cid:107) π R ( Q η ( X ) − Q η ( X )) (cid:107) s − + (cid:107) D R π R ( Q η ( X ) − Q η ( X )) (cid:107) s − for n = 1and ≺ (cid:107) π R ( Q η ( X ) − Q η ( X )) (cid:107) s − + (cid:107) d R π R ( Q η ( X ) − Q η ( X )) (cid:107) s − for n > . Evoking Lemma 4.6.2 and Lemma 4.6.3 then yields the estimate of the first term we need: (cid:107)| ( π ⊥ ◦ Φ( X ) − π ⊥ ◦ Φ( X )) (cid:107)| s ≺ (cid:107) ( X − X ) (cid:107) s − ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s )+ (cid:107) ( X − X ) (cid:107) s ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) . We estimate the second term in (4.6.6) as follows. We first claim that Φ( Y − Y ) = π ⊥ ◦ Φ( Y − Y ).To see this, use the definition Ψ( X ) = Φ − ◦ π ◦ Φ( X ) for any vector field X and the linearity ofthe map π ◦ Φ to write π ◦ Φ( Y i ) in the form π ◦ Φ( Y i ) = π ◦ Φ(Ψ( X i ) − X i ) = π ◦ Φ(Φ − ◦ π ◦ Φ( X i )) − π ◦ Φ( X i ) = 0 . It follows that π ◦ Φ( Y − Y ) = 0; hence, π ⊥ ◦ Φ( Y − Y ) = Φ( Y − Y ). Next, since Ψ( X ) =Φ − ◦ π ◦ Φ( X ), we have π ⊥ ◦ Φ ◦ Ψ( X i ) = π ⊥ π ◦ Φ( X i ) = 0 . Combining these two identities shows the second term in (4.6.6) can be writtenΦ( Y − Y ) + ( π ⊥ ◦ Φ( X ) − π ⊥ ◦ Φ( X ))= π ⊥ { Φ( Y − Y ) + (Φ( X ) − Φ(Ψ( X )) − Φ( X ) + Φ(Ψ( X )) } . We now show that(4.6.7) π ⊥ { Φ( Y − Y ) + (Φ( X ) − Φ(Ψ( X )) − Φ( X ) + Φ(Ψ( X )) } = L ( Q η ( Y − Y ) − ( Q η (Ψ( X )) − Q η (Ψ( X ))) + ( Q η ( X ) − Q η ( X )))where L is the linear operator defined by L ( φ ) := (cid:40) { ( n + 1) G R δ R ⊕ ( G R D ∗ R D R + H R ) } ◦ π R ( φ ) , for n = 1 { ( n + 1) G R δ R ⊕ ( n G R δ R d R + H R ) } ◦ π R ( φ ) , for n > φ = η to the vector fields Y − Y , X i and Ψ( X i ) to obtainthe three expansions F ∗ X i η = η + L X i η + Q η ( X i ) F ∗ Ψ( X i ) η = η + L Ψ( X i ) η + Q η (Ψ( X i )) F ∗ ( Y − Y ) η = L Y η − L Y η + Q η ( Y − Y ) . Substituting these expressions into Equation (4.6.5) and collecting terms reveals that the termsinvolving the Lie derivative of η cancel to yield the identity (4.6.7). Thus, by the triangle inequality,to estimate the second term in (4.6.6), we need only estimate each term in the sum ||| L ( Q η ( Y − Y )) ||| s + ||| L ( Q η (Ψ( X )) − Q η (Ψ( X ))) ||| s + ||| L ( Q η ( X ) − Q η ( X )) ||| s For each term, we use the estimates ||| L ( φ ) ||| s ≺ (cid:40) (cid:107) π R φ (cid:107) s − + (cid:107) D R π R φ (cid:107) s − + (cid:107) H R π R φ (cid:107) s , for n = 1 (cid:107) π R φ (cid:107) s − + (cid:107) d R π R φ (cid:107) s − + (cid:107) H R π R φ (cid:107) s , for n > L and employing Lemmas 4.6.2and 4.6.3 shows that ||| L ( Q η ( Y − Y )) ||| s ≺ ( (cid:107) Y − Y (cid:107) s ) (cid:107) Y − Y (cid:107) s − , ||| L ( Q η (Ψ( X )) − Q η (Ψ( X ))) ||| s ≺ ( (cid:107) Ψ( X ) (cid:107) s + (cid:107) Ψ( X ) (cid:107) s ) · (cid:107) Ψ( X ) − Ψ( X ) (cid:107) s − + ( (cid:107) Ψ( X ) (cid:107) s − + (cid:107) Ψ( X ) (cid:107) s − ) · (cid:107) Ψ( X ) − Ψ( X ) (cid:107) s , ONTACT DIFFEOMORPHISMS 41 and ||| L ( Q η ( X ) − Q η ( X )) ||| s ≺ ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) · (cid:107) X − X (cid:107) s − + ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) · (cid:107) X − X (cid:107) s To conclude the estimate of (4.6.6), note that since Ψ is C , we can replace Ψ( X i ) by X i and Y − Y by X − X everywhere in the first two of the previous three inequalities. (cid:3) References [Arn] V. I. Arnold, Mathematical Methods of Classical Mechanics , Springer-Verlag (1978), New York, Heidelberg,Berlin.[B] J. Bland, Contact geometry and CR-structures on S , Acta Math. (1994), 1–49.[BD2] J. Bland and T. Duchamp, Anisotropic Estimates for Sub-elliptic Operators , Sci. in China, Ser. A: Mathematics (Science Press/ Springer-Verlag) (2008), 509–522.[BD3] J. Bland and T. Duchamp, The space of Cauchy-Riemann structures on 3-D compact contact manifolds ,(preprint).[Bi] O. Biquard, Metriques autoduales sur la boule , Invent. math. (2002), 545-607.[CL] J.-H. Cheng and J.M. Lee, A local slice theorem for 3-dimensional CR structures , Amer. J. Math. (1995),1249–1298.[E] D. Ebin, The manifold of Riemannian metrics , in “Global Analysis”, eds. S.S. Chern and S. Smale, Proc.Symp. in Pure Math., Amer. Math. Soc. Vol XV (1970), 11–40.[EM] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid , Ann. Math. (1070) 102–163.[FS] G. B. Folland and E. M. Stein, Estimates for the ¯ ∂ b complex and analysis on the Heisenberg group , Comm. onPure and Applied Math. (1974), 429–522.[G] J. Gray, Some global properties of contact structures , Ann. of Math. (1959), 421–450.[Ham] R. Hamilton, The inverse function theorem of Nash and Moser , Bull. AMS (1982) 65–222.[O1] H. Omori, On the group of diffeomorphisms on a compact manifold , in “Global Analysis”, eds. S.S. Chern andS. Smale, Proc. Symp. in Pure Math., Amer. Math. Soc. Vol XV (1970), 167–183.[O2] H. Omori, Infinite dimensional lie transformations groups , Lecture Notes in Mathematics, vol. 427, Springer-Verlag, Berlin, Heidelberg, New York, 1974.[Pal] R. Palais, Foundations of Global Non-Linear Analysis , W.A. Benjamin, Inc. (1968), New York.[R] M. Rumin, Formes differentielles sur les varietes de contact , J. Diff. Geom. (1994), 281–330. John Bland, University of TorontoTom Duchamp, University of Washington E-mail address : [email protected] E-mail address ::