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Home , Pseudoholomorphic curve, Symplectic manifold, Symplectomorphism

LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES

WEIMIN CHEN, UMASS, SPRING 07

In this lecture we give a general introduction to the basic concepts and some of the fundamental problems in symplectic /, where along the way various examples are also given for the purpose of illustration. We will often give statements without proofs, which means that their proof is either beyond the scope of this course or will be treated more systematically in later lectures.

1. Symplectic Throughout we will assume that M is a C∞-smooth without boundary (unless specific mention is made to the contrary). Very often, M will also be closed (i.e., compact). Definition 1.1. A symplectic structure on a smooth manifold M is a 2-form ω ∈ Ω2(M), which is (1) nondegenerate, and (2) closed (i.e. dω = 0). (Recall that a 2-form ω ∈ Ω2(M) is said to be nondegenerate if for every point p ∈ M, ω(u, v) = 0 for all u ∈ TpM implies v ∈ TpM equals 0.) The pair (M, ω) is called a . Before we discuss examples of symplectic manifolds, we shall first derive some im- mediate consequences of a symplectic structure. (1) The nondegeneracy condition on ω is equivalent to the condition that M has an even dimension 2n and the top wedge product ωn ≡ ω ∧ ω · · · ∧ ω is nowhere vanishing on M, i.e., ωn is a . In particular, M must be orientable, and is canonically oriented by ωn. The nondegeneracy condition is also equivalent to the condition that M is almost complex, i.e., there exists an endomorphism J of TM such that J 2 = −Id. The latter is homotopy theoretic in nature as it means that the tangent bundle TM as a SO(2n)-bundle can be lifted to a U(n)-bundle under the natural homomorphism U(n) → SO(2n). (More systematic discussion in Lecture 2.) Note that this has nothing to do with the closedness of ω. (2) The closedness of ω is a very important, nontrivial geometric or analytical condition. For now, let us simply observe that ω defines a deRham cohomology class [ω] ∈ H2(M), which must be nonzero when M is closed. In fact in this case, [ω]n ≡ [ω] ∪ [ω] · · · ∪ [ω] ∈ H2n(M) n R n must be nonzero, since [ω] ([M]) = M ω 6= 0. As an example, this shows that for n ≥ 2, the spheres S2n admit no symplectic structures even though they are 1 2 WEIMIN CHEN, UMASS, SPRING 07

all even dimensional and orientable (because S2n is closed and H2(S2n) = 0 when n ≥ 2). Of course, the failure may simply be caused by nonexistence of nondegenerate 2-forms (or equivalently, nonexistence of almost complex struc- tures) which is homotopy theoretic in nature, rather than by the failure of closedness of the 2-forms which is geometric or analytical in nature. This is almost true, except for the case of S6. Example 1.2. (Calabi, 1958). S6 admits an almost complex structure. 7 An important fact here is that there is a vector product × in R (related to Cayley numbers), which is bilinear and skew symmetric, and is related to the standard inner product h, i by the following rules: hu × v, wi = hu, v × wi and (u × v) × w + u × (v × w) = 2hu, wiv − hv, wiu − hv, uiw. The key properties we shall need here are (1) u × v is orthogonal to both u, v, and (2) u × (u × v) = −v if u is a unit vector, which can be easily derived from the above rules. 6 7 6 Now for any x ∈ S ⊂ R , let ν(x) be the unit outward normal vector of S at x. We define 6 Jxu = ν(x) × u, ∀u ∈ TxS . 6 2 2 Then Jxu ∈ TxS because it is orthogonal to ν(x), and Jx = −Id because Jxu = ν(x) × (ν(x) × u) = −u as ν(x) is a unit vector. Note: It is an open question as whether S6 is a ; the almost complex structure defined above is NOT integrable. Here is a fundamental question in symplectic topology. Problem 1.3. (Existence of symplectic structures). Suppose M is an of dimension 2n where n ≥ 2, and moreover, when M is closed, there exists a cohomology class a ∈ H2(M) such that 0 6= an ∈ H2n(M). Does M admit a symplectic structure? If not, what additional assumptions do we need? Note that for the case of n = 1, i.e., M is a , the problem is trivial (and the answer is yes). The case where M is open (without boundary) was solved affirmatively by M. Gromov, which shows that in this case the existence of a symplectic structure is a homopopy theoretic (the so-called “soft”) problem. Theorem 1.4. (Gromov, 1969). Suppose M is an open manifold. Then every non- degenerate 2-form on M (as long as it exists) is homotopic through nondegenerate 2-forms on M to a symplectic structure. The case where M is closed turns out to be much harder. Gromov’s technique (the so-called “h-principle”) does not work in this case, but for a long time no one has been able to provide a nagetive example until C. Taubes’ work on the Seiberg-Witten invari- ants of symplectic 4-manifolds. This work reveals an important connection between the symplectic topology and differential topology of 4-manifolds. LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 3

2 2 2 Theorem 1.5. (Taubes, 1994). The 4-manifold M = CP #CP #CP is almost com- plex and has a cohomology class a ∈ H2(M) such that a2 6= 0, however, M does not admit any symplectic structure. Now we give some natural examples of symplectic manifolds. Example 1.6. (Euclidean Spaces). The most basic examples are the Euclidean spaces 2n R equipped with the standard symplectic structure

ω0 = dx1 ∧ dy1 + dx2 ∧ dy2 + ··· + dxn ∧ dyn. 2n n Show that with R ≡ C under zj = xj + iyj, j = 1, 2 ··· , n, n i X ω = dz ∧ dz¯ . 0 2 j j j=1 Example 1.7. (Cotangent Bundles). Let L be a smooth manifold of dimension n and M ≡ T ∗L be the . There is a canonical symplectic structure on M which has the form ω = −dλ. The 1-form λ is defined as follows. Let π : M ≡ T ∗L → L be the natural projection. ∗ ∗ ∗ ∗ Then for any v ∈ M ≡ T L, we have π : Tπ(v)L → Tv M. With this understood, λ ∗ ∗ is defined by setting its value at v to be λ(v) ≡ π (v). (Note that v ∈ Tπ(v)L, so that ∗ ∗ λ(v) ≡ π (v) ∈ Tv M.) Let q1, ··· , qn be local coordinates on L, and p1, ··· , pn be the corresponding coor- Pn dinates on the fibers, i.e, if a cotangent vector v = j=1 pjdqj, then v has coordinates p1, ··· , pn on the fiber. Together q1, ··· , qn and p1, ··· , pn form a system of local ∗ Pn coordinates on M ≡ T L. In the above local coordinates, we claim λ = j=1 pjdqj. Accepting it momentarily, we see immediately that ω ≡ −dλ is a symplectic structure, Pn because ω = j=1 dqj ∧ dpj in these coordinates. Pn Pn To see the claim λ = j=1 pjdqj, recall that the value of λ at v = j=1 pjdqj equals ∗ Pn ∗ Pn π (v) = j=1 pjπ (dqj) = j=1 pjd(π ◦ qj). Since q1, ··· , qn are regarded as local ∗ Pn coordinates on M ≡ T L, qj = π ◦ qj, and with this understood, v = j=1 pjdqj has Pn coordinates q1, ··· , qn, p1, ··· , pn. This shows that λ = j=1 pjdqj. It is interesting to check that λ is characterized by the following property: for any 1-form σ on L, which can be regarded as a of the cotangent bundle π : M ≡ T ∗L → L, one has σ∗λ = σ. We remark that cotangent bundles form a fundamental class of symplectic man- ifolds. These are the phase spaces in classical mechanics, with coordinates q and p corresponding to position and momentum. Example 1.8. (Orientable Surfaces). Let M be a 2-dimensional orientable manifold, and let ω be any volume form on M. Then (M, ω) is a symplectic manifold because in this case ω is automatically nondegenerate and closed. Example 1.9. (K¨ahlerManifolds). Let M be a complex manifold of dimension n, and let h be a Hermitian metric on M. Then in a local complex coordinates z1, z2, ··· , zn, Pn h may be written as h = j,k=1 hjk¯dzj ⊗ dz¯k, where (hjk¯) is a n × n Hermitian ¯ Pn matrix, i.e., hjk¯ = hk¯j. The associated 2-form to h is ω = i j,k=1 hjk¯dzj ∧ dz¯k. With 4 WEIMIN CHEN, UMASS, SPRING 07 this understood, h is called a K¨ahlermetric if ω is closed. ω is clearly nondegenerate, hence the underlying real manifold, also denoted by M, is a 2n-dimensional symplectic manifold with a symplectic structure ω. (Note that the nondegeneracy condition on ω follows from the identity n n ω = i det(hjk¯) dz1 ∧ dz¯1 ∧ · · · ∧ dzn ∧ dz¯n. Verify the identity.) An important aspect of is its connection with almost K¨ahler geometry. One can regard M as a 2n-dimensional real manifold equipped with an (integrable) almost complex structure J. In this context h can be regarded as a J-invariant complex symmetric bilinear form on the complexification of the almost complex vector bundle TM, and ω and h are related by h(·, ·) = ω(·,J(·)). n The complex projective spaces CP form a fundamental class of K¨ahlermanifolds. n There is a canonical K¨ahlermetric on CP , called the Fubini-Study metric. In terms n of the homogeneous coordinates z0, z1, ··· , zn of CP , the associated K¨ahlerform of the Fubini-Study metric is n i X X ω0 = Pn 2 (¯zjzjdzk ∧ dz¯k − z¯jzkdzj ∧ dz¯k). 2π · ( zκz¯κ) κ=0 k=0 j6=k R n (We point out that ω0 is normalized such that n ω0 = 1.) Note that a complex n CP of CP is naturally a K¨ahlermanifold with the induced metric. Example 1.10. (Product of Symplectic Manifolds). Given two symplectic manifolds (M1, ω1), (M2, ω2), the product M1 ×M2 is also symplectic with symplectic structures ∗ ∗ π1ω1 ⊕ π2(±ω2). Here πi : M1 × M2 → Mi, i = 1, 2, and we canonically identity ∗ ∗ ∗ ∗ ∗ T (M1 × M2) with π1(T M1) ⊕ π2(T M2). A of a symplectic manifold (M, ω) is a diffeomorphism ψ ∈ Diff(M) which preserves the symplectic structure ω = ψ∗ω. Note that a symplectomorphism is particularly a volume-preserving diffeomorphism (hence is necessarily an orientation-preserving diffeomorphism), as one has ψ∗(ωn) = (ψ∗ω)n = ωn. There is an abundance of on a symplectic mani- fold. In fact, the group of symplectomorphisms, denoted by Symp(M, ω) or simply by Symp(M), is an infinite dimensional , in contrast with the finite dimension- ality of the group of a Riemannian metric. It has been an open question as whether symplectomorphisms differ from volume- preserving diffeomorphisms, until in 1985 M. Gromov published the revolutionary paper on pseudoholomorphic curves in symplectic manifolds, where he proved the following celebrated non-squeezing result. 2n 2n Theorem 1.11. (Gromov, 1985). One can not squeeze the unit ball B (1) ⊂ R 2 2n−2 2n into a “long and thin” cylinder B (r) × R ⊂ R (where r < 1) via an embedding 2n which preserves the symplectic structure ω0 on R , however, one can always squeeze n the ball into the “long and thin” cylinder if only the volume form ω0 is required to be preserved. LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 5

The nondegeneracy condition of a symplectic structure ω gives rise to the following canonical isomorphism TM → T ∗M : X 7→ ι(X)ω = ω(X, ·). A vector field X is called symplectic if ι(X)ω is closed. The next result is one consequence of the closedness of a symplectic structure, which shows that when M is closed, the set of symplectic vector fields form the of the group Symp(M).

Proposition 1.12. Let M be a closed manifold. If t 7→ ψt ∈ Diff(M) is a smooth family of diffeomorphisms generated by a family of vector fields Xt via d ψ = X ◦ ψ , ψ = id, dt t t t 0 then ψt ∈ Symp(M) for every t iff Xt is a symplectic vector field for every t. Proof. Recall Cartan’s formula for the

LX ω = ι(X)dω + d(ι(X)ω). Now the closedness of ω, i.e., dω = 0, implies that d ψ∗ω = ψ∗(L ω) = ψ∗(d(ι(X )ω), dt t t Xt t t which vanishes if and only if ι(Xt)ω is closed. This proves that ψt ∈ Symp(M) for every t iff Xt is a symplectic vector field for every t.  There is a simple way to obtain symplectic vector fields, and therefore to obtain sym- plectomorphisms, which shows their abundance. Let H be a smooth function (which has a compact support if M is not closed). There is a vector field XH canonically associated to H by the following equation

ι(XH )ω = dH. t The flow ψH on M generated by XH , i.e., d ψt = X ◦ ψt , dt H H H is called a Hamiltonian flow. Note that XH is a symplectic vector field as ι(XH ), t being exact, is closed. Thus ψH is a symplectomorphism for each t. The function H is called the Hamiltonian function and the vector field XH is called the Hamiltonian vector field. More generally, a symplectomorphism ψ ∈ Symp(M) is called Hamiltonian if there exists a smooth family of ψt ∈ Symp(M), t ∈ [0, 1], with ψ0 = id, ψ1 = ψ, such that the corresponding (time-dependent) symplectic vector field Xt generating ψt is Hamiltonian, i.e., the 1-form ι(Xt)ω is exact for each t and has the form ι(Xt)ω = dHt for a time-dependent smooth function Ht on M. The function Ht is called a time-dependent Hamiltonian function, and ψt is called an Hamiltonian isotopy. 6 WEIMIN CHEN, UMASS, SPRING 07

1 Example 1.13. (A Hamiltonian S -Action). Consider the symplectic manifold (M, ω) 3 where M is the unit sphere in R , i.e. 2 2 2 M = {(x1, x2, x3)|x1 + x2 + x3 = 1}, 3 and ω is the area form induced from the embedding M ⊂ R . We will show that the Hamiltonian flow on M associated to the “height” function

H(x1, x2, x3) ≡ x3 1 is the S -action on M defined by the rotation about the x3-axis. To see this, we change x1, x2 into polar coordinates r, θ, and note that in terms of 3 r, θ, x3 the volume form of R is rdr ∧ dθ ∧ dx3. On the unit sphere M, r = 1 so that the restriction of the volume form on M is v = dr ∧ dθ ∧ dx3. This implies that ∂ ω = ι( )v = dθ ∧ dx . ∂r 3 Recall that the Hamiltonian vector field XH is defined by ι(XH )ω = dH. It follows ∂ that for H = x3, XH = ∂θ , and therefore the associated Hamiltonian flow is given by the rotation about the x3-axis. 1 The following question is a basic one concerning Hamiltonian S -actions. When dim M = 2 or 4, the question is completely understood, and for the case of dim M = 6, partial results have been obtained. 1 Question 1.14. If a symplectic S -action has a fixed point, is it necessarily Hamil- 1 tonian? Is there a characterization of Hamiltonian S -actions in terms of fixed points data? Now we go back to the discussion of examples of symplectic manifolds. Note that the examples we gave earlier of closed symplectic manifolds are all coming from K¨ahler manifolds. (The example of orientable surfaces is K¨ahlerbecause every almost complex structure in dimension 2 is integrable.) The following problem has been a basic scheme in symplectic topology, much of which has become known only in the last ten years. Problem 1.15. (Symplectic > K¨ahler).How much larger is the category of symplectic manifolds than the category of K¨ahlermanifolds? Consider the following way of constructing symplectic manifolds. Suppose (M, ω) is a symplectic manifold and a discrete group G acts on M via symplectomorphisms, i.e., for every g ∈ G, the diffeomorphism g : M → M is an element of Symp(M, ω). Suppose furthermore that the action of G is free, and hence the quotient M/G is again a smooth manifold. Then ω descends to a symplectic structure on the quotient manifold M/G, with which M/G naturally becomes a symplectic manifold. For a typical example, 2n Pn consider R with the standard symplectic structure ω0 = j=1 dxj ∧ dyj, and the 2n free action by translation of G = Z the integral lattice. The quotient is the (2n)- 2n dimensional torus T , with a standard symplectic structure. The following example is also of this sort, which gives the first example of a closed, symplectic but non-K¨ahlermanifold. (The example was known to Kodaira in the 1950s and was rediscovered in the 1970s by Thurston.) LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 7

2 2 Example 1.16. (A Non-K¨ahlerManifold). Consider the group G = Z × Z with the noncommutative group operation   0 0 0 0 1 j2 (j , k ) ◦ (j, k) = (j + j ,A 0 k + k ),A = , j j 0 1 2 4 where j = (j1, j2) ∈ Z , and similarly for k. This group acts on R via 4 G → Diff(R ):(j, k) 7→ ρjk, where ρjk(x, y) = (x + j, Ajy + k). One can verify easily that the action is free and preserves the symplectic structure

ω = dx1 ∧ dx2 + dy1 ∧ dy2. 4 Hence the quotient M = R /G is a symplectic manifold which is easily seen to be closed. 3 We will show that H1(M; Z) = Z so that M is not a K¨ahlermanifold. (Recall that by the Hodge theory the odd dimensional Betti numbers of a closed K¨ahlermanifold must be even.) To see this, one first verifies that the commutator subgroup [G, G], −1 −1 which consists of elements of the form aba b , a, b ∈ G, equals 0 ⊕ 0 ⊕ Z ⊕ 0. Then note that π1(M) = G and H1(M; Z) = π1(M)/[π1(M), π1(M)]. Hence H1(M; Z) = 3 G/[G, G] = Z . This implies that the first Betti number of M equals 3, and that M is not K¨ahler. 2 2 1 Topologically, M is a nontrivial T bundle over T , or more precisely, M = S × N 2 1 where N is the nontrivial T bundle over S defined as follows. Let x, y be the standard 2 2 coordinates on T . Then N = [0, 1] × T / ∼, where (0, x, y) ∼ (1, x + y, y). 2. of Symplectic Manifolds Let (M, ω) be a symplectic manifold, and let Q ⊂ M be a submanifold of M. Note that the tangent bundle TQ is a sub-bundle of TM|Q. We set ω [ TQ ≡ {u ∈ TMq|ω(u, v) = 0 ∀v ∈ TQq}, q∈Q which is also a sub-bundle of TM|Q. We say Q is isotropic if TQ ⊂ TQω, coisotropic if TQω ⊂ TQ, symplectic if TQ ∩ TQω = {0}, and Lagrangian if TQ = TQω. Amongst these four types of submanifolds of a symplectic manifold, Lagrangian submanifolds form the most important class, which is what we shall concentrate in this section. Note that a Lagrangian submanifold in a 2n-dimensional symplectic manifold has dimension n. Example 2.1. (Totally real submanifolds in a K¨ahlermanifold). Let M be a K¨ahler manifold of complex dimension n. A real n-dimensional submanifold Q is called totally real if for every point q ∈ Q there is a local holomorphic complex coordinates centered n n at q within which Q is identified with the real part R ⊂ C . Check that a totally real submanifold in a K¨ahlermanifold is Lagrangian with respect to the K¨ahlerform. When n = 1, every symplectic manifold is K¨ahler,and every real 1-dimensional submanifold is Lagrangian and totally real. For higher dimensional examples, suppose M is a N complex submanifold of CP , which is K¨ahlerunder the induced metric. Then the 8 WEIMIN CHEN, UMASS, SPRING 07

fixed-point set of the anti-holomorphic involution zi 7→ z¯i in M, if nonempty, is a totally real submanifold of M. 2n Example 2.2. (Lagrangian submanifolds in (R , ω0)). The n-dimensional spaces defined by xj = constant, j = 1, ··· , n, or yj = constant, j = 1, ··· , n, are Lagrangian 2n submanifolds in (R , ω0). For a different type of examples, consider the n-torus n 2n 2n 2 T ⊂ R , where we think (R , ω0) as a product of symplectic manifold (R , ω0) and T n ≡ S1 × · · · × S1, 1 2 2 2 where the j-th copy of S is the unit circle {xj + yj = 1} in the j-th copy of R . The torus T n is Lagrangian because (1) any 1-dimensional manifold in a 2-dimensional symplectic manifold is Lagrangian, and (2) the product of Lagrangian submanifolds in a product symplectic manifold is again Lagrangian. Example 2.3. (Lagrangian submanifolds and symplectomorphisms). Let (M, ω) be any symplectic manifold. Then in the product symplectic manifold (M ×M, ω×(−ω)), the diagonal ∆ ≡ {(x, x) ∈ M × M|x ∈ M} is Lagrangian. More generally, for any ψ ∈ Symp(M, ω), the gragh of ψ in M × M, gragh(ψ) ≡ {(x, ψ(x)) ∈ M × M|x ∈ M}, is a Lagrangian submanifold. Example 2.4. (Lagrangian submanifolds in cotangent bundles). Let M ≡ T ∗L be the cotangent bundle of L equipped with the canonical symplectic structure ω = −dλ. P P ∗ (In local coordinates λ = j pjdqj and ω = j dqj ∧ dpj.) Clearly, the fibers of T L ∗ (defined by qj = constant, ∀j) and the zero section L ⊂ T L (defined by pj = 0, ∀j) are Lagrangian submanifolds. Next we consider submanifolds Qσ in M which is the graph of a 1-form σ on L, ∗ regarded as a smooth section of T L. Since Qσ is of half dimension of M, it follows that Qσ is a Lagrangian submanifold if and only if the pull-back of ω to Qσ equals 0, which is equivalent to the condition that the pull-back σ∗ω = 0. But σ∗ω = σ∗(−dλ) = −d(σ∗λ) = −dσ, which implies that Qσ is Lagrangian iff σ is closed. One of the fundamental questions in symplectic topology concerns the intersection of Lagrangian submanifolds. In order to motivate this question, we consider the in- tersection of Qσ for a nonzero closed 1-form σ with the zero section Q0 = L. Observe that the set of intersection points of Qσ and L is precisely the set of zeroes of σ on L. With this understood, one should distinguish the following two cases: (1) σ = df is exact, (2) σ is not exact. In the first case, Qσ is called an exact Lagrangian sub- manifold, and the function f : L → R is called a generating function of Qσ. Now the basic observation is that on a closed manifold exact 1-forms σ = df have more zeroes (which correspond to the critical points of the function f) than closed 1-forms 2 in general. For instance, on the 2-torus T , any smooth function has at least 3 critical points and any Morse function has at least 4 critical points (which equals the sum of 2 the Betti numbers), while there are closed 1-forms on T which are nowhere vanishing. LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 9

Lemma 2.5. Suppose σ is a closed 1-form on L and Qσ is the corresponding La- ∗ ∗ grangian submanifold in T L. Let ψt ∈ Symp(T L), t ∈ [0, 1], be an isotopy of symplectomorphisms such that ψ0 = id and ψ1(L) = Qσ. Then σ is exact if ψt is an Hamiltonian isotopy.

Proof. Let Xt be the corresponding time-dependent symplectic vector field. Then d ψ∗λ = ψ∗(L λ) = ψ∗(ι(X )dλ + d(ι(X )λ)), dt t t Xt t t t ∗ which implies that ψt λ − λ is exact for each t if ψt is an Hamiltonian isotopy, i.e., ι(Xt)ω = ι(Xt)dλ is exact for every t. The lemma follows by observing that, since ψ1(L) = Qσ, ∗ ∗ ∗ σ = σ λ = i (ψ1λ − λ), where i : L → T ∗L is the zero section. 

Problem 2.6. (Arnold’s conjecture on Lagrangian intersections). Let L0,L1 be com- pact Lagrangian submanifolds of a symplectic manifold (M, ω). Moreover, suppose that ψt ∈ Symp(M, ω), t ∈ [0, 1], is an Hamiltonian isotopy such that ψ0 = id and L1 = ψ1(L0). Then L0 and L1 must have at least as many intersection points as a function on L0 must have critical points. Let (M, ω) be a closed symplectic manifold, and let ψ ∈ Symp(M, ω). Then as we have seen earlier that both the diagonal ∆ and the graph of ψ are Lagrangian sub- manifolds of the product (M × M, ω × (−ω)). Moreover, observe that the intersection points of ∆ and the graph of ψ are exactly the fixed points of ψ in M, i.e., x ∈ M such that ψ(x) = x. Now suppose ψt is an Hamiltonian isotopy on M. Then Ψt ≡ (id, ψt) is an Hamiltonian isotopy on M × M, and Ψt(∆) is the graph of ψt for every t. Problem 2.7. (Arnold’s conjecture on symplectic fixed points). Let ψ : M → M be an Hamiltonian symplectomorphism of a closed symplectic manifold (M, ω). Then ψ must have at least as many fixed points as a function on M must have critical points. Note that Arnold’s conjecture on symplectic fixed points is trivially true when the Hamiltonian symplectomorphism ψ is generated by a time-independent Hamiltonian H on M. Indeed, in this case, the symplectic fixed points are precisely the critical points of the Hamiltonian function H. We remark that Arnold’s conjectures have been one of the focal points of much of the recent developement in symplectic topology. The celebrated theory was initially introduced to tackle these conjectures!

3. Contact Manifolds Contact topology is the odd-dimensional analogue of symplectic topology. While contact topology is an interesting subject of its own right, the interplay between sym- plectic topology and contact topology has become a trend in recent years. Definition 3.1. Let N be a (2n + 1)-dimensional manifold, and let ξ ⊂ TN be a hyperplane field, which is defined by a 1-form α, i.e., ξ = ker α. ξ is called a contact 10 WEIMIN CHEN, UMASS, SPRING 07 structure if dα is nondegenerate when restricted to ξ, or equivalently, α ∧ (dα)n 6= 0. The 1-form α is called a contact form associated to ξ, and the pair (N, ξ) is called a contact manifold. Remark 3.2. (1) Recall that the Frobenius integrability theorem says that a hyper- plane field ξ = ker α is integrable if and only if α ∧ dα = 0. So a contact structure may be thought of as a maximally nonintegrable distribution. (2) Note that α ∧ (dα)n is a volume form on N, in particular, N is orientable, and is canonically oriented by α ∧ (dα)n. On the other hand, dα is nondegenerate on ξ, so that ξ is canonically oriented by (dα)n. These together determines an orientation of the quotient bundle T N/ξ. In other words, the contact structure ξ is both oriented and co-oriented. (3) Suppose α0 is another 1-form which defines ξ, i.e., ξ = ker α0. Then α0 = fα where f is a nowhere vanishing function. One can easily check that α0 ∧ (dα0)n = f n+1α ∧ (dα)n 6= 0. Hence α0 is another contact form associated to ξ, and ξ being a contact structure is independent of the choice of its defining 1-form. We emphasize that the contact structure is the hyperplane field ξ not the contact form α. It determines the contact forms up to a nonzero factor f. We usually assume that f > 0 so that the relevant orientations of the contact structure defined through the contact forms are compatible. Definition 3.3. Let (N, ξ) be a contact manifold of dimension 2n + 1, and let L be a n-dimensional submanifold of N. L is called Legendrian if TL ⊂ ξ|L. In a certain sense Legendrian submanifolds are contact analogues of Lagrangian submanifolds. Suppose α is any contact form associated to a contact structure ξ, and suppose L is a Legendrian submanifold. Then dα = 0 on TL. To see this, let X,Y be any vector fields on L. Then [X,Y ] is also a vector field on L. Since TL ⊂ ξ|L, we have α(X) = α(Y ) = α([X,Y ]) = 0, which implies that dα(X,Y ) = 0.

Example 3.4. (Euclidean spaces). Let x1, ··· , xn, y1, ··· , yn, z be the coordinates 2n+1 on R . Then the 1-form n X α0 = dz − yjdxj j=1 2n+1 is a contact form on R . Let’s visualize the corresponding contact structure ξ0 for 3 3 the case of R . Let x, y, z be the coordinates on R . Then α0 = dz − ydx. It is easily seen that [ ξ0 = {a∂y + b(∂x + y∂z)|a, b ∈ R}. 3 (x,y,z)∈R Example 3.5. (1-jet bundles). Let L be a n-dimensional manifold, and let N ≡ ∗ T L × R be the 1-jet bundle over L. Then α = dz − λ LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 11 is a contact form on N, where z is the coordinate on the R factor and λ is the ∗ canonical 1-form on the T L factor. Note that for any smooth function f : L → R, the submanifold Lf ≡ {(x, df(x), f(x))|x ∈ L} ⊂ N is Legendrian. Example 3.6. (Circle bundles). Let N be a circle bundle over Σ. Then for any 1 connection 1-form A ∈ Ω (N; iR) of the bundle, FA = dA descents to a 2-form on Σ such that the first of the circle bundle π : N → Σ is represented by iFA/2π. Now suppose that the first Chern class of the circle bundle π : N → Σ is represented by a 2-form ω which is a symplectic structure on Σ. Then one can choose ∗ a connection 1-form A such that FA = −2πiπ ω. We set 1 α ≡ A. 2π Then α is a contact 1-form on N with dα = π∗ω. One basic example of the above 2n+1 n construction is given by the Hopf fibration π : S → CP , where the first Chern n class is represented by the K¨ahlerform of the Fubini-Study metric on CP . Let (N, ξ) be a contact manifold and α be a contact form associated to the contact structure ξ. A diffeomorphism ψ ∈ Diff(N) is called a contactomorphism if ψ preserves the oriented hyperplane field ξ, or equivalently, if there is a smooth function h ∗ h such that ψ α = e α.A contact isotopy is a smooth family ψt of contactomorphisms ∗ ht such that ψt α = e α for a smooth family of functions ht. A vector field X on a contact manifold (N, ξ) is called a contact vector field if LX α = gα for a contact form α associated to ξ, where g is a smooth function on N.

Consider a time-dependent contact vector field Xt, such that LXt α = gtα, and the smooth family of diffeomorphisms ψt generated by Xt, i.e., d ψ = X ◦ ψ , ψ = id. dt t t t 0 One can check easily that ψt is a contact isotopy. In fact, since d ψ∗α = ψ∗L α = ψ∗g α, dt t t Xt t t

∗ ht R t ∗ one has ψt α = e α where ht = 0 ψs gsds. Conversely, if ψt is a contact isotopy with ∗ ht −1 ∗ d ψt α = e α, then LXt α = gtα where gt = (ψt ) dt ht. Given any contact form α, there is a canonical contact vector field Xα, called a

Reeb vector field, which has the property LXα α = 0. Xα is uniquely determined by the conditions ι(Xα)dα = 0 and ι(Xα)α = 1. The dynamical system generated by a Reeb vector field is called a Reeb dynamics. More generally, fixing a contact form α, there is a 1-1 correspondence between contact vector fields and smooth functions as follows:

ι(XH )α = −H and ι(XH )dα = dH − (ι(Xα)dH)α, where H is a smooth function and XH is the corresponding contact vector field. Note that Xα corresponds to H ≡ −1. 12 WEIMIN CHEN, UMASS, SPRING 07

We close this section with a discussion on the interaction between contact topology and symplectic topology. Recall that symplectic geometry is the geometry on which Hamiltonian mechan- ics rests. One of the fundamental problems in concerns the existence of periodic solutions, i.e., periodic orbits of the corresponding Hamiltonian dynamical system. More concretely, suppose (M, ω) is a symplectic manifold and H is a Hamiltonian function on M such that each level surface H−1(c) is compact. Let XH be the corresponding Hamiltonian vector field, defined by

ι(XH )ω = dH.

Note that the Hamiltonian dynamical system generated by XH preserves the level surfaces of H as seen easily from the fact XH (H) = ω(XH ,XH ) = 0. In particular, the flow ψt defined by d ψ = X ◦ ψ , ψ = id dt t H t 0 exists for t ∈ (−∞, ∞) since each level surface H−1(c) is compact. A periodic orbit of the Hamiltonian dynamical system is a map γ : R → M which satisfies d γ(t) = X (γ(t)), ∀t ∈ , and γ(T ) = γ(0) for some T > 0. dt H R Note that every periodic orbit is contained in some level surface of H. Given any contact manifold (N, ξ) with a contact form α, there is a canonical symplectic structure on R × N defined as follows: let t be the coordinate on the R t factor, then the 2-form d(e α) is a symplectic structure on R × N. The symplectic t manifold (R×N, d(e α)) is called the symplectization of (N, α). Note that the vector t field ∂t on R × N has the property that L∂t ω = ω where ω ≡ d(e α) is the canonical symplectic structure on R × N. Such a vector field is called a Liouville vector field. Let (M, ω) be a symplectic manifold, and let N ⊂ M be a compact hypersurface. We say that N is of contact type if there exists a contact form α on N such that a neighborhood of N in M can be symplectically identified with a neighborhhod of 2n−1 {0} × N in the symplectization of (N, α). For example, the unit sphere S in 2n (R , ω0) is a hypersurface of contact type with contact form n 1 X α = (x dy − y dx ). 0 2 j j j j j=1 Problem 3.7. (Weinstein’s conjecture on Hamiltonian dynamics). Suppose a level −1 surface H (c) is of contact type in (M, ω). Then there exists a periodic orbit of XH contained in H−1(c). Note that suppose H−1(c) is of contact type and α is the contact form on H−1(c) with respect to which a neighborhood of H−1(c) in (M, ω) is identified symplectically with a neighborhhod of {0} × H−1(c) in the symplectization. Then it is easily seen that ω|H−1(c) = dα. In particular, XH is in the null direction of dα, i.e., ι(XH )dα = 0. Notice that the periodic orbits of XH are in 1-1 correspondence with the periodic −1 orbits of the Reeb vector field Xα on H (c). LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 13

Problem 3.8. (Weinstein’s conjecture on Reeb dynamics). Let N be a compact contact manifod with contact form α. Then there exists at least one periodic orbit of the Reeb vector field Xα. For another aspect of contact topology in symplectic geometry, let’s consider sym- plectic structures on a manifold M with boundary. Suppose M is almost complex, or equivalently, M admits a nondegenerate 2-form. Then by Gromov’s theorem the interior of M admits a symplectic structure. Of course, in general one has no a priori knowledge about the behavior of the symplectic structure near the boundary ∂M. Let M be a manifold with boundary, and let ω be a symplectic structure on M. We say (M, ω) is a symplectic manifold with contact boundary if ∂M is of contact type and a neighborhood of ∂M in (M, ω) is symplectically identified with (−, 0]×∂M 2n 2n in the symplectization for some  > 0. For example, the unit ball B (1) ⊂ R with the standard symplectic structure ω0 is a symplectic manifold with contact boundary. While Gromov’s work shows that open manifolds have a rather trivial symplec- tic topology, closed symplectic manifolds exhibit many rigidity properties, discovered mainly through his powerful theory of pseudoholomorphic curves. Symplectic man- ifolds with contact boundary resemble in many aspects closed symplectic manifolds, because the theory can be properly extended to this context. Symplectic geometry also provides a useful tool in the study of contact topology, especially in dimension 3. A compact contact manifold (N, ξ) is called symplecti- cally fillable if (N, ξ) can be realized as the boundary of a symplectic manifold with 3 contact boundary. For example, the contact structure on S obtained via the Hopf 3 1 fibration S → CP as described in Example 3.6 is symplectically fillable; it can be re- 4 alized as the boundary of the symplectic manifold with contact boundary (B (1), ω0). Symplectically fillable contact structures on a 3-manifold belong to a class of contact structures, called tight contact structures, which exhibit many rigidity properties.

References [1] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Mono- graphs, 2nd edition, Oxford Univ. Press, 1998.