1 Contact structures and Reeb vector fields

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1 1 Contact structures and Reeb vector fields Definition 1.1. Let M be a smooth (2n + 1)-dimensional manifold. A contact structure on M is a maximally non-integrable hyperplane field ξ T M. Suppose that there exists a global 1-form α such that ξ = ker α (then ξ is called coorientable). Then the non-integrability condition is equivalent to α (dα) n > 0, i.e. the form α (dα) n is a volume form on M. The pair (M, ξ) is called a contact manifold and α is called a contact form. Example 1.2. Consider the euclidean space R 2n+1 and a 1-form Then dα = α = dz + x i dy i. dx i dy i α (dα) n = dz dx 1 dy 1... dx n dy n, thus α is a contact form. The contact structure determined by α is called the standard contact structure. Remark 1.3. The form α (dα) n is a volume form consequently M is necessarily orientable. Remark 1.4. Any other contact form, which determines the same contact structure, has to be of the form fα for some non-vanishing f C (M). Lemma 1.5. Suppose that dim M = 3 and let ξ = ker α be a hyperplane field on M. Then ξ is a contact structure if and only if for all pointwise linearly independent vector fileds X, Y ξ we have [X, Y ] / ξ. Proof. Let X, Y ξ be two vector fields on M. Then dα(x, Y ) = X(α(Y )) Y (α(x)) α([x, Y ]) = α([x, Y ]). Contact condition for α is satisfied if dα ker α 0. Thus α([x, Y ]) 0 as claimed. Example 1.6. Consider R 3. Then α = dz + xdy and ξ is spanned by vector fields x and y x z. [ x, y x z ] = z. Definition 1.7. The Reeb vector field R of a contact form α is defined by the following equations { dα(r, ) 0, α(r) 1. 1

2 Remark 1.8. If a contact form α is fixed, then the Reeb vector field is uniquely defined. Definition 1.9. A Liouville vector field Y on a symplectic manifold (W 2n+2, ω) is a vector field satisfying L Y ω = ω. Remark By the Cartan formula L Y β = d(i Y β) + i Y dβ, thus is β is closed, then L Y β = d(i Y β). Lemma Let Y be a Liouville vector field on a symplectic manifold (W, ω). Suppose that M is a codimension one submanifold of W transverse to Y. Then α = i Y ω is a contact form on M. Proof. From the definition of the Liouville vector field it follows that α (dα) n = i Y ω (d(i y ω)) n = i Y ω ω n = 1 n + 1 i Y ω n+1. Consequently α (dα) n is a volume form on any hypersurface transverse to Y. Example Let (M, ξ) be a contact manifold and let α be a contact form. Let W = M R and ω = d(e t α). The contact condition implies that ω is a symplectic form on W. Additionally t is a Liouville vector field transverse to M. Thus every contact manifold can be obtained as a hypersurface of a symplectic manifold transverse to some Liouville vector field. Symplectic manifold (W, ω) is called symplectization of (M, α). 2 Gray stability and the Moser trick Lemma 2.1. Let ω t be a smooth 1-parameter family of k-forms on some manifold M. Let ψ t be an isotopy of M. Define a vector field X t by the equality X t ψ t = ψ t, i.e. ψ t is a flow of X t. Then the following equality holds d dt (ψ t ω t ) = ψt ( ω t + L Xt ω t ). Proposition 2.2. Let ξ t, t [0, 1] be a smooth family of contact structures ona closed manifold M. Then there exists an isotopy ψ t of M such that T ψ t (ξ 0 ) = ξ t. Proof. Let α t be a family of contact 1-forms corresponding to ξ t. Our aim is to find an isotopy of M satisfying ψ t α t = λ t α 0 (1) 2

3 for some 1-parameter family of positive smooth functions λ t. Let s assume that ψ t is a flow of a time-dependent vector field X t. differentiating (1) and using lemma 2.1 we obtain ψy ( t + L Xt α t ) = λ t α 0 ψt λ ( t + d(α t (X t )) + i Xt dα t ) = t ψt α t = ψt (µ t α t ), λ t Now λ where µ t = t λ t ψt 1. Each ψ t is a diffeomorphism, thus the above equality is equivalent with the following t + d(α t (X t )) + i Xt dα t = µ t α t. (2) Assume now that X t ξ t. Then (2) implies that t + i Xt dα t = µ t α t. (3) Plugging in the Reeb vector field R t = R αt yields the following If we define µ t by the equality 4, then t (R t ) = µ t. (4) ( t µ t α t ) (R t ) 0. (5) The non-degeneracy of dα t ξt impllies that there exists a unique X t ξ t such that (3) is satisfied. Consequently if ψ t is a flow of X t, which is globally defined since M is closed, then ψ t is an isotopy with the desired property. Theorem 2.3 (Darboux Theorem). Let (M, α) be a contact manifold. Given p M, there exists a coordinate chart around p such that in the local coordinates the following equality holds α = dz + x i dy i. Proof. Without loss of generality we can ussume that M = R 2n+1 and p = 0. Using linear changes of coordinates we can arrange that at 0 xi 0, yi 0 ξ 0 and following equalities hold Define α( z ) 0 = 1, (i z dα) 0 = 0, (dα) 0 = (dx i dy i ) 0. α 0 = dz + x i dy i, α t = (1 t)α 0 + tα. 3

4 Notice that α t 0 = α 0 0. Now we would like to apply the Moser trick to find an isotopy ψ t such that ψ t α t = α 0. (6) Differentiating 6 yields ψt ( t + L Xt α t ) = 0, (7) t + d(α(x t )) + i Xt dα t = 0. (8) Vector field X t can be written as a sum X t = H t R t + Y t, where Y t ker α t and H t is a smooth function. Then (7) becomes Pluging in the Reeb vector field gives t + dh t + i Yt dα t = 0. (9) t (R t ) + dh t (R t ) = 0. This equation can be solved locally for H t. Then the proof proceeds analogously as the proof of the theorem Neighbourhood theorems Definition 3.1. A knot k in a contact 3-manifold (M 3, ξ) is called legendrian if for every point p k we have T p k ξ p. Theorem 3.2. Let k (M 3, ξ) be a legendrian knot. Then there are coordinates (θ, x, y) in a neighbourhood of k = S 1 {0} S 1 R 2 = ν(k, M) such that ξ ν(k,m) = ker(cos θdx sin θdy). Here ν(k, M) denotes the tubular neighbourhood of k in M. Proof. Without loss of generality we can assume that M = S 1 R 2 and k = S 1 {0}. Let α be a contact form on M and define a 1-form α 0 = ker(cos θdx sin θdy). Assume that θ is a coordinate on S 1 {0}. Define an automorphism φ: ν(s 1 {0}, M) ν(s 1 {0}, M) as follows. Choose vector fields X 0, X such that α(x) = α 0 (X 0 ) = 0 and dα( θ, X) = dα 0 ( θ, X 0 ) = 1. Then φ acts as follows R α0 R α X 0 X. This definition determines φ uniquely. Now let τ : ν(s 1 {0}, M) S 1 R 2 be a tubular map. This τ is a diffeomorphism on a neighbourhood of the zero section of ν(s 1 {0}, M). Furthermore τ S1 {0} = id, Dτ S1 {0} = id. 4

5 The composition τ φ τ 1 is a diffeomorphism of a neighbourhood of S 1 {0}, thus α 0 and α 1 = (τ φ τ 1 ) α are contact forms on a neighbourhood of S 1 {0} with α 0 T S 1 {0} = α 1, dα 0 T S 1 {0} = dα 1. To finish the proof it is sufficient to apply Moser trick to α t = (1 t)α 0 + tα 1. 4 Contact vector fields and Hamiltonian functions Let (W, ω) be a symplectic manifold. Choose a smmoth function H C (W ). It defines a unique vector field X h by the relation dh = ω(x H, ). Vector field X H is called a hamiltonian vector field corresponding to the Hamiltonian H. The flow of X H preserves the symplectic form ω, because L XH ω = d(i XH ω) + i XH dω = d(i XH ω) = d 2 H = 0. Now our goal is to give a similar construction in contact geometry. Definition 4.1. A vector field X on a contact manifold (M, ξ) is called a contact vector field if its flow preserves the contact structure ξ. Example 4.2. Let M = S 1 R 2 and α = cos θdx sin θdy. Then α is a contact form on M. Consider a vector field X = x x + y y. We have L X α = i X dα + d(i X α) = = i X ( sin θdθ dx cos θdθ dy) + d(x cos θ y sin θ) = = x sin θdθ + y cos θdθ + cos θdx sin θdy x sin θdθ y cos θdθ = = α. Thus ψ t α = e t α, so T ψ t (ξ) = ξ. Consequently X is a contact vector field. Remark 4.3. In general if X is a contact vector field for ξ = ker α, then L X α = µα for some function µ. Proposition 4.4. Let (M, ξ) be a contact manifold and let ξ = ker α. Then there exists a bijection {Contact vector fields on M} C (M). The correspondence is given by the following formulas X H X = α(x) C (M) C (M) H X H, 5

6 where X H is a vector field defined by the following two equations { α(xh ) = H, i XH dα = dh(r α )α dh. Proof. Let s check first that X H is a contact vector field. L XH α = i XH dα + d(i XH α) = = dh(r α )α dh + dh = dh(r α )α. Thus X H is a contact vector field indeed. In order to check that the map given in the theorem is bijective one can compute the compositions of both maps. For the first composition we obviously have H XH = H. To check the other composition notice that i XHX dα = dh X (R α )α dh X = L X d(i X α). Using Cartan formula we obtain the following equality Additionally Equalities 10 and 11 imply that X = X HX. i XHX dα = i X dα. (10) α(x HX ) = α(x). (11) 5 Isotopy extension theorem Theorem 5.1. Let (M, ξ) be a contact 3-manifold and let J t : S 1 M be an isotopy of legendrian embeddings. Then there is a contact isotopy ψ t of M such that ψ t j 0 = j t. Proof. Define X t on j t (S 1 ) by X t j t = d dt j t. Let H t = α(x t ) be a contact Hamiltonian corresponding to X t. In order to solve the isotopy extension problem, we have to find a compactly supported function H t : M R, which extends H t defined on j t (S 1 ), then X Ht is a contact vector field extending X t. The flow of X Ht gives the desired isotopy. Recall that X t = X Ht is defined in terms of H t by α( X t ) = H t, i Xt dα = d X t (R α )α d H t. 6

7 The X t and H t agree with X t and H t on j t (S 1 ), respectively. Hence we need α(x t ) = H t, i Xt dα = d H t (R α )α d H t (12) along j t (S 1 ). The first condition states that H t = H t on j t (S 1 ). Since j t is a legendrian embedding, so T (j t (S 1 )) ξ. Hence the second condition implies that for any v T j t (S 1 ) thus 0 = dα(x t, v) + d H t (v) = dα(x t, v) + dh t (v), j t (dα(x t, ) + dh t ) = j t (i Xt dα + d(i Xt α)) = = j t (i Xt dα) + d(j t i Xt α) = d dt (j t α) = 0, since j t is the isotopy of contact embeddings so the above condition is satisfied. To sum up, to define H t it is enough to define it on j t (S 1 ) and define its differential d H t along j t (S 1 ). The first part is done by the first condition from 12. To fullfill the second condition define d H t (R) = 0 and for v ker α the value of d H t is defined by the second condition from 12. 7

THE EXISTENCE PROBLEM

THE EXISTENCE PROBLEM Contact Geometry in High Dimensions Emmanuel Giroux CNRS ENS Lyon AIM May 21, 2012 Contact forms A contact form on a manifold V is a non-vanishing 1-form α whose differential dα at