LECTURE3:LOCALNORMALFORMSCONTENTS11.Isotopy42.Moser's trick63.Darbouxstyletheorems1.ISOTOPY Some backgrounds on the Lie derivative.Let M be a smooth manifold, and X a smooth vector field on M. If X iscomplete, i.e. the integral curve p generated by X starting from p is defined overR for all p.Then X generates a flow (ot) on M, i.e..For each t, ot : M → M isa diffeomorphism,.For each t, s, pt o ps= pt+s.where t is defined explicitly by t(p)=p(t).We remark that if X is not completeone can still define a local flow near any given point on M, which is enough for mostof what follows.Now let α be any k-form on M. Then the Lie derivative of α with respect to Xisdefinedtobethek-formd I(1)Cxa:ota.dt lt=0Note that according the group law,dd(2)a=Lxa.diadss≥0For the Lie derivative, thefollowing formula is very useful and is known as theCartan's magic formula:(3)Lxa=dixa+ixda,where tx is the contraction operator.This can be proved via three steps: (1) checkthe formula holds for functions, (2) check both sides commutes with the differentiald, (3) check both sides are derivatives for the algebra (2*(M),^), e,g, Cx(α β) =(Cxa)Aβ+αALxβetc.1
LECTURE 3: LOCAL NORMAL FORMS Contents 1. Isotopy 1 2. Moser’s trick 4 3. Darboux style theorems 6 1. Isotopy ¶ Some backgrounds on the Lie derivative. Let M be a smooth manifold, and X a smooth vector field on M. If X is complete, i.e. the integral curve γp generated by X starting from p is defined over R for all p. Then X generates a flow {φt} on M, i.e. • For each t, φt : M → M is a diffeomorphism, • For each t, s, φt ◦ φs = φt+s. where φt is defined explicitly by φt(p) = γp(t). We remark that if X is not complete, one can still define a local flow near any given point on M, which is enough for most of what follows. Now let α be any k-form on M. Then the Lie derivative of α with respect to X is defined to be the k-form (1) LXα = d dt t=0 φ ∗ tα. Note that according the group law, (2) d dtφ ∗ tα = d ds s=0 φ ∗ tφ ∗ sα = φ ∗ tLXα. For the Lie derivative, the following formula is very useful and is known as the Cartan’s magic formula: (3) LXα = dιXα + ιXdα, where ιX is the contraction operator. This can be proved via three steps: (1) check the formula holds for functions, (2) check both sides commutes with the differential d, (3) check both sides are derivatives for the algebra (Ω∗ (M), ∧), e,g, LX(α ∧ β) = (LXα) ∧ β + α ∧ LXβ etc. 1
2LECTURE3:LOCALNORMALFORMSTIsotopies.Definition 1.1.A smooth family ot:M→M of diffeomorphismswith po=Id iscalled an isotopy.For each isotopy @t one can construct a time-dependent vector field Xt viaddtt= Xi(o),or in other words,dXt(p) =ps(p-t(p).dtConversely, given any compactly-supported time dependent vector field Xt, one canconstruct an isotopy pt so that the previous relation holds. We will call this isotopythe flow generated by Xt. When X, is not compactly-supported, such a “flow" stillexistslocallyneareachpoint.One can extend the equation (2) to isotopies:Lemma 1.2.Let X, be a time-dependent vector field with flow ot.Then for Va E2k(M),dga=dCxaSketch. One need to (1) check the formula holds for functions, (2) check both sidescommuteswiththedifferential d.(3)checkbothsidesarederivativesforthealgebra口(2*(M), Λ).It followsProposition 1.3.Let Qt be a smooth family of k-forms.Thenddatlda=(cxat+Proof. According to chain rule, for any smooth function f(r, y) of two variables,ddd(t,t)=lr=tf(r,t) +ly=tf(t,g).2drdyApply this to our case, we getddddadfay=prat+Xotdeoatdedydt口Remark. In what follows we will denote dar by &(t)
2 LECTURE 3: LOCAL NORMAL FORMS ¶ Isotopies. Definition 1.1. A smooth family φt : M → M of diffeomorphisms with φ0 = Id is called an isotopy. For each isotopy φt one can construct a time-dependent vector field Xt via d dtφt = Xt(φt), or in other words, Xt(p) = d dt s=t ρs(ρ−t(p)). Conversely, given any compactly-supported time dependent vector field Xt , one can construct an isotopy φt so that the previous relation holds. We will call this isotopy the flow generated by Xt . When Xt is not compactly-supported, such a “flow” still exists locally near each point. One can extend the equation (2) to isotopies: Lemma 1.2. Let Xt be a time-dependent vector field with flow φt. Then for ∀α ∈ Ω k (M), d dtφ ∗ tα = φ ∗ tLXtα. Sketch. One need to (1) check the formula holds for functions, (2) check both sides commutes with the differential d, (3) check both sides are derivatives for the algebra (Ω∗ (M), ∧). It follows Proposition 1.3. Let αt be a smooth family of k-forms. Then d dtφ ∗ tαt = φ ∗ t LXtαt + dαt dt . Proof. According to chain rule, for any smooth function f(x, y) of two variables, d dtf(t, t) = d dx|x=tf(x, t) + d dy |y=tf(t, y). Apply this to our case, we get d dtφ ∗ tαt = d dx x=t φ ∗ xαt + d dy y=t φ ∗ tαy = φ ∗ t LXtαt + dαt dt . Remark. In what follows we will denote dαt dt by ˙α(t).
3LECTURE3:LOCALNORMALFORMSHomotopyformula.Let X, be a time-dependent vector field whose flow t exists for 0 <t <1. Forany Q E *(M) one defines Qt(α) = tx,(,a) and letQ(α) =:Qt(a)dt.Then Q is a map from *(M) to k-1(M) which is obviously linear.Theorem1.4 (Homotopyformula).Φa-Q=dQ(α)+Q(da)Proof.One hasd%0ia = Lx(ota) = dx:(ota) + xd(eta) = d(a) + Q(da),whichimplies) dt = dQ(α) + Q(da),pa-口More generally, let f,g : Mi→ M, be two smooth maps that are homotopic, i.e.there exists a smooth map F : Mi × R → M2 so that F(p,0) = f(p) and F(p, 1) =g(p) for all p E Mi. Then there exists a homotopy operator Q : 2*(M2)-→ k-1(M))sothat(4)g*- f*=dQ+QdTo see this, one just apply the previous theorem to the manifold W =M ×x R. Inthis case the vector field is complete with flow Φt(p, a) = (p,a +t). So one gets alinear homotopy map Q : *(W) -→ 2k-1(W) such thatΦ - = dQ +Qd.On the other hand wehave f=F otand g=F opr ot,where t :Mi W is theinclusion. So one getg*-f*=t*Φ*F*-t*F*=t*(dQ+Qd)F*=dl*QF*+t*QdF*The conclusion follows if one take Q= t*QF*.As a consequence, we seeCorollary 1.5.Let t:N M be a submanifold,α E (M) a closed k-form onM such that t*a = O. Then one can find and a neighborhoodu of N in M and a(k - 1)-form βek-1(u) with β = 0 on N such that α = dβ on u.Remark.Thefact "β=0 on N"isin thesenseof (k-1)-form on M,thusismuchstronger than t*β = 0
LECTURE 3: LOCAL NORMAL FORMS 3 ¶ Homotopy formula. Let Xt be a time-dependent vector field whose flow φt exists for 0 ≤ t ≤ 1. For any α ∈ Ω k (M) one defines Qt(α) = ιXt (φ ∗ tα) and let Q(α) = Z 1 0 Qt(α)dt. Then Q is a map from Ωk (M) to Ωk−1 (M) which is obviously linear. Theorem 1.4 (Homotopy formula). φ ∗ 1α − α = dQ(α) + Q(dα). Proof. One has d dtφ ∗ tα = LXt (φ ∗ tα) = dιXt (φ ∗ tα) + ιXtd(φ ∗ tα) = d(Qt(α)) + Qt(dα), which implies φ ∗ tα − α = Z 1 0 d dtφ ∗ tα dt = dQ(α) + Q(dα). More generally, let f, g : M1 → M2 be two smooth maps that are homotopic, i.e. there exists a smooth map F : M1 × R → M2 so that F(p, 0) = f(p) and F(p, 1) = g(p) for all p ∈ M1. Then there exists a homotopy operator Q˜ : Ωk (M2) → Ω k−1 (M1) so that (4) g ∗ − f ∗ = dQ˜ + Qd. ˜ To see this, one just apply the previous theorem to the manifold W = M1 × R. In this case the vector field ∂ ∂t is complete with flow φt(p, a) = (p, a + t). So one gets a linear homotopy map Q : Ωk (W) → Ω k−1 (W) such that φ ∗ 1 − φ ∗ 0 = dQ + Qd. On the other hand we have f = F ◦ ι and g = F ◦ φ1 ◦ ι, where ι : M1 ,→ W is the inclusion. So one get g ∗ − f ∗ = ι ∗φ ∗ 1F ∗ − ι ∗F ∗ = ι ∗ (dQ + Qd)F ∗ = dι∗QF∗ + ι ∗QdF∗ . The conclusion follows if one take Q˜ = ι ∗QF∗ . As a consequence, we see Corollary 1.5. Let ι : N ,→ M be a submanifold, α ∈ Ω k (M) a closed k-form on M such that ι ∗α = 0. Then one can find and a neighborhood U of N in M and a (k − 1)-form β ∈ Ω k−1 (U) with β = 0 on N such that α = dβ on U. Remark. The fact “β = 0 on N” is in the sense of (k − 1)-form on M, thus is much stronger than ι ∗β = 0.
4LECTURE3:LOCALNORMALFORMSProof.By choice of Riemannian metric and its exponential map, one can find aneighborhood u of X in M and a smooth retract of onto X, that is, a one-parameter family of smooth maps rt : u -→ u and a smooth map π : u -→ X suchthat ri=Id, ro=t o π and rt ot =t.Applying the homotopy formula (4) one getsα-元*t*α = (dQ+Qd)a.Since aα is closed and iα = O, we see α = dQα. So one only need to take β = Qa口Itremainsto checkβ=OonN.2.MOSER'S TRICKT Moser's theorem.This following method was first used by J. Moser in a very short paper in 1965.which turned out to be very useful in many situations, and thus is widely known asMoser'stricknow.Suppose we have two k-forms Qo and αi on a smooth manifold M and we aretrying to find a diffeomorphism : M -→ M such that Φ*ai = αo. Moser's trick isto construct as the time-1 flow map of a time-dependent vector field Xt on M. Infact, Moser's trick does much more:for a smooth family of k-forms, at, connectingQo and ai, one try to find a time-dependent vector field Xt on M so that its flowΦt: M -→M satisfies, for all 0≤t≤1,(5)$tat = ao.To solve the equation (5), one only need to solved0=at =d(a + Cx,).Inserting the Cartan's magic formula,the equation to be solved becomes(6)Qt+dix,at+tx,dat=0.The last equation is much easier to solve in many cases. As an illustration of thismethod, we proveTheorem 2.1 (Moser). Let M be compact and ao,α1 two volume forms on M.Then there erists a diffeomorphism Φ:M →M such that *ai =ao if and only ifJMQ0=JMQ1.Proof. If such a diffeomorphism exists, then obviously/ g*α1=Qo =Q11J(M)JNJAJMConversely suppose Jμ Qo = JM Q1, i.e. JM(aα1 - Qo) = 0. Then[α1 - Qo] = 0 E HdeRham(M)
4 LECTURE 3: LOCAL NORMAL FORMS Proof. By choice of Riemannian metric and its exponential map, one can find a neighborhood U of X in M and a smooth retract of ι onto X, that is, a oneparameter family of smooth maps rt : U → U and a smooth map π : U → X such that r1 = Id, r0 = ι ◦ π and rt ◦ ι = ι. Applying the homotopy formula (4) one gets α − π ∗ ι ∗α = (dQ˜ + Qd˜ )α. Since α is closed and ι ∗α = 0, we see α = dQα˜ . So one only need to take β = Qα˜ . It remains to check β = 0 on N. 2. Moser’s trick ¶ Moser’s theorem. This following method was first used by J. Moser in a very short paper in 1965, which turned out to be very useful in many situations, and thus is widely known as Moser’s trick now. Suppose we have two k-forms α0 and α1 on a smooth manifold M and we are trying to find a diffeomorphism φ : M → M such that φ ∗α1 = α0. Moser’s trick is to construct φ as the time-1 flow map of a time-dependent vector field Xt on M. In fact, Moser’s trick does much more: for a smooth family of k-forms, αt , connecting α0 and α1, one try to find a time-dependent vector field Xt on M so that its flow φt : M → M satisfies, for all 0 ≤ t ≤ 1, (5) φ ∗ tαt = α0. To solve the equation (5), one only need to solve 0 = d dtφ ∗ tαt = φ ∗ t ( ˙αt + LXtαt). Inserting the Cartan’s magic formula, the equation to be solved becomes (6) ˙αt + dιXtαt + ιXtdαt = 0. The last equation is much easier to solve in many cases. As an illustration of this method, we prove Theorem 2.1 (Moser). Let M be compact and α0, α1 two volume forms on M. Then there exists a diffeomorphism φ : M → M such that φ ∗α1 = α0 if and only if R M α0 = R M α1. Proof. If such a diffeomorphism exists, then obviously Z M α0 = Z M φ ∗α1 = Z φ(M) α1 = Z M α1. Conversely suppose R M α0 = R M α1, i.e. R M (α1 − α0) = 0. Then [α1 − α0] = 0 ∈ H n deRham(M),
LECTURE3:LOCALNORMALFORMS5i.e. there exists β n-1(M) so thatQ1 - Qo = dβ.Now letat = (1 - t)ao +ta1.Then at is a family of volume forms connecting Qo and ai, and &t =Qi -Qo.Wewant to find an isotopy pt so that tot = Qo, which implies the theorem. Accordingto Moser's trick, it is enough to solve the equation (6), which, in our case, becomes0=Qt+dx,at+ix,dat=d(β+tx,at).This is always solvable, because one can always find a vector field X, solving theequationβ+tx,at = 0,口since ot's are volume forms.Remark.In fact Moserproved more:thereexists a smooth family of diffeomorphismsΦt and a smooth family of at such that Φ*at = ao.Classification of 2-dimensional compact symplectic manifolds.As an application of Moser't theorem, we haveTheorem 2.2 (Classification of compact symplectic surfaces). Let (Mi,wi) and(M2,w2) be two closed 2-dimensional symplectic manifolds. Then they are symplec-tomorphic if and only if they have the same genus and the same symplectic area.Proof.This follows from thefact that two smooth compact surfaces are diffeomorphic口if and only if they have the same genus together with Moser's theorem.Remark.This is no such classification theorem for dimensions ≥4.↑ Deformation of symplectic structure in the same cohomology class.As another application of Moser's trick, one can prove that a deformation in thesame de Rham cohomology class will not give us any new symplectic structure, i.e.Theorem 2.3.Let M be compact and wt = wo + dβt a smooth family of symplecticforms on M.Then there erists a smooth family of diffeomorphisms pt : M M sothato,wt=wo.Proof.Repeat Moser's argument as before.Now Moser's equation (6) becomesd(βt +tx,wt) = 0,and thus it is enough to find find a vector field Xt solvingβt+lxwt=0,which is always solvable because of the non-degeneracy of the symplectic form
LECTURE 3: LOCAL NORMAL FORMS 5 i.e. there exists β ∈ Ω n−1 (M) so that α1 − α0 = dβ. Now let αt = (1 − t)α0 + tα1. Then αt is a family of volume forms connecting α0 and α1, and ˙αt = α1 − α0. We want to find an isotopy φt so that φ ∗ tαt = α0, which implies the theorem. According to Moser’s trick, it is enough to solve the equation (6), which, in our case, becomes 0 = ˙αt + dιXtαt + ιXtdαt = d(β + ιXtαt). This is always solvable, because one can always find a vector field Xt solving the equation β + ιXtαt = 0, since αt ’s are volume forms. Remark. In fact Moser proved more: there exists a smooth family of diffeomorphisms φt and a smooth family of αt such that φ ∗ tαt = α0. ¶ Classification of 2-dimensional compact symplectic manifolds. As an application of Moser’t theorem, we have Theorem 2.2 (Classification of compact symplectic surfaces). Let (M1, ω1) and (M2, ω2) be two closed 2-dimensional symplectic manifolds. Then they are symplectomorphic if and only if they have the same genus and the same symplectic area. Proof. This follows from the fact that two smooth compact surfaces are diffeomorphic if and only if they have the same genus together with Moser’s theorem. Remark. This is no such classification theorem for dimensions ≥ 4. ¶ Deformation of symplectic structure in the same cohomology class. As another application of Moser’s trick, one can prove that a deformation in the same de Rham cohomology class will not give us any new symplectic structure, i.e. Theorem 2.3. Let M be compact and ωt = ω0 + dβt a smooth family of symplectic forms on M. Then there exists a smooth family of diffeomorphisms φt : M → M so that φ ∗ tωt = ω0. Proof. Repeat Moser’s argument as before. Now Moser’s equation (6) becomes d(β˙ t + ιXtωt) = 0, and thus it is enough to find find a vector field Xt solving β˙ t + ιXtωt = 0, which is always solvable because of the non-degeneracy of the symplectic form.