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
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).
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.
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),
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.
6 LECTURE 3: LOCAL NORMAL FORMS 3. Darboux style theorems ¶ Weinstein’s proof of Darboux’s theorem. Before we prove Darboux theorem, we first prove Theorem 3.1 (Weinstein). Let M be a smooth manifold and i : N ,→ M a compact submanifold. Let ω0 and ω1 be two symplectic forms on M such that ω0|N = ω1|X. Then there exist neighborhoods U0 and U1 of N in M and a smooth map φ : U0 → U1 such that φ|N = Id and φ ∗ω1 = ω0. Proof. Let ωt = (1 − t)ω0 + tω1. Since ωt = ω0 on N and N is compact, one can find a tubular neighborhood U of N so that ωt is symplectic on U for all 0 ≤ t ≤ 1. According to the corollary of the homotopy formula above, there exists a 1-form α on U with α|N = 0 such that ˙ωt = ω1 − ω0 = dα on U. Again we solve the equation ιXtωt + α = 0 to get a vector field Xt on U. Since α = 0 on N, we see Xt = 0 on N. So we may shrink U to a neighborhood U0 of N so that the flow of Xt is defined for 0 ≤ t ≤ 1 on U0. Now set φ to be the time-1 map of the flow of Xt on U0 and set U1 = φ(U0). As a consequence, we get Theorem 3.2 (Darboux’s theorem). Let (M, ω) be a symplectic manifold of dimension 2n. Then for any p ∈ M, there exists a coordinate patch (U, x1, · · · , xn, ξ1, · · · , ξn) centered at p such that on U, ω = Xdxi ∧ dξi . Proof. Pick any symplectic basis {x 0 1 . · · · , x0 n , ξ0 1 , · · · , ξ0 n} for the symplectic vector space (TpM, ωp), and extend it to a coordinate system in a neighborhood U 0 of p. On U 0 one has two symplectic forms: the given one ω0 = ω, and a new one ω1 = Pdx0 i ∧ dξ0 i . Now apply the previous theorem with X = {p} and M = U 0 , we can find neighborhood U0 and U1 of p in U 0 and a diffeomorphism ϕ : U0 → U1 so that ϕ(p) = p and ϕ ∗ ( Xdx0 i ∧ dξ0 i ) = ω. To complete the proof we only need to set xi = ϕ ∗ (x 0 i ) and ξi = ϕ ∗ (ξ 0 i ). ¶ Lagrangian neighborhood theorem. Theorem 3.3 (Weinstein). Let M be a smooth manifold of dimension 2n and ω1, ω2 two symplectic forms on M. Let ι : X ,→ M be a submanifold of M which is Lagrangian with respect to both ω1 and ω2. Then there exists neighborhoods U0 and U1 of X in M and a diffeomorphism ϕ : U0 → U1 with ϕ|X = Id such that φ ∗ω1 = ω0. Proof. Student presentation.
LECTURE 3: LOCAL NORMAL FORMS 7 As a consequence we can show that near a Lagrangian submanifold the symplectic manifold “looks like” the cotangent bundle! Theorem 3.4 (Tubular neighborhood theorem for Lagrangian). Let (M, ω) be a symplectic manifold and ι : X ,→ M a Lagrangian submanifold. Then there exists a neighborhood U0 of X in T ∗X, a neighborhood U of X in M and a diffeomorphism ϕ : U0 → U with ϕ|X = Id so that ϕ ∗ω = ω0. Proof. We will need • Let M be a smooth manifold and ι : X ,→ M be a submanifold. Definition 3.5. The normal bundle NX of X in M is a vector bundle over X whose fiber at x ∈ X is NxX = TxM/TxX. • If M is symplectic and X a Lagrangian submanifold, then NX is canonically identified with T ∗X: At each point x ∈ X the symplectic form gives a canonical non-degenerate pairing NxX × TxX → R,([v], u) 7→ ωx(v, u), using which one gets a canonical identification of NxX with T ∗ xX. • The standard tubular neighborhood theorem: Theorem 3.6. Let M be a n dimensional manifold and ι : X ,→ M a kdimensional submanifold. Then there exist a neighborhood U0 of X in NX, a neighborhood U of X in M and a diffeomorphism ψ : U0 → U so that ψ|X = Id. Back to the the theorem. By the facts above, we can find a neighborhood U0 of X in T ∗X, a neighborhood U of X in M and a diffeomorphism ψ : U0 → U such that ψ|X = Id. Now on the manifold U0 one has two symplectic forms: ω0 = the canonical symplectic form on T ∗X, and ω1 = ψ ∗ω. Moreover, X is a Lagrangian submanifold with respect to both symplectic forms. Now the theorem follows from theorem 3.3. Remark. The theorem extends further to isotropic submanifolds.