LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES

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1 LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES WEIMIN CHEN, UMASS, SPRING 07 In this lecture we give a general introuction to the basic concepts an some of the funamental problems in symplectic geometry/topology, 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 beyon the scope of this course or will be treate more systematically in later lectures. 1. Symplectic Manifols Throughout we will assume that M is a C -smooth manifol without bounary (unless specific mention is mae to the contrary). Very often, M will also be close (i.e., compact). Definition 1.1. A symplectic structure on a smooth manifol M is a 2-form ω Ω 2 (M), which is (1) nonegenerate, an (2) close (i.e. ω = 0). (Recall that a 2-form ω Ω 2 (M) is sai to be nonegenerate if for every point p M, ω(u, v) = 0 for all u T p M implies v T p M equals 0.) The pair (M, ω) is calle a symplectic manifol. Before we iscuss examples of symplectic manifols, we shall first erive some immeiate consequences of a symplectic structure. (1) The nonegeneracy conition on ω is equivalent to the conition that M has an even imension 2n an the top wege prouct ω n ω ω ω is nowhere vanishing on M, i.e., ω n is a volume form. In particular, M must be orientable, an is canonically oriente by ω n. The nonegeneracy conition is also equivalent to the conition that M is almost complex, i.e., there exists an enomorphism J of T M such that J 2 = I. The latter is homotopy theoretic in nature as it means that the tangent bunle T M as a SO(2n)-bunle can be lifte to a U(n)-bunle uner the natural homomorphism U(n) SO(2n). (More systematic iscussion in Lecture 2.) Note that this has nothing to o with the closeness of ω. (2) The closeness of ω is a very important, nontrivial geometric or analytical conition. For now, let us simply observe that ω efines a erham cohomology class [ω] H 2 (M), which must be nonzero when M is close. In fact in this case, [ω] n [ω] [ω] [ω] H 2n (M) must be nonzero, since [ω] n ([M]) = M ωn 0. As an example, this shows that for n 2, the spheres S 2n amit no symplectic structures even though they are 1

2 2 WEIMIN CHEN, UMASS, SPRING 07 all even imensional an orientable (because S 2n is close an H 2 (S 2n ) = 0 when n 2). Of course, the failure may simply be cause by nonexistence of nonegenerate 2-forms (or equivalently, nonexistence of almost complex structures) which is homotopy theoretic in nature, rather than by the failure of closeness of the 2-forms which is geometric or analytical in nature. This is almost true, except for the case of S 6. Example 1.2. (Calabi, 1958). S 6 amits an almost complex structure. An important fact here is that there is a vector prouct in R 7 (relate to Cayley numbers), which is bilinear an skew symmetric, an is relate to the stanar inner prouct, by the following rules: u v, w = u, v w an (u v) w + u (v w) = 2 u, w v v, w u v, u w. The key properties we shall nee here are (1) u v is orthogonal to both u, v, an (2) u (u v) = v if u is a unit vector, which can be easily erive from the above rules. Now for any x S 6 R 7, let ν(x) be the unit outwar normal vector of S 6 at x. We efine J x u = ν(x) u, u T x S 6. Then J x u T x S 6 because it is orthogonal to ν(x), an Jx 2 = I because Jxu 2 = ν(x) (ν(x) u) = u as ν(x) is a unit vector. Note: It is an open question as whether S 6 is a complex manifol; the almost complex structure efine above is NOT integrable. Here is a funamental question in symplectic topology. Problem 1.3. (Existence of symplectic structures). Suppose M is an almost complex manifol of imension 2n where n 2, an moreover, when M is close, there exists a cohomology class a H 2 (M) such that 0 a n H 2n (M). Does M amit a symplectic structure? If not, what aitional assumptions o we nee? Note that for the case of n = 1, i.e., M is a Riemann surface, the problem is trivial (an the answer is yes). The case where M is open (without bounary) was solve affirmatively by M. Gromov, which shows that in this case the existence of a symplectic structure is a homopopy theoretic (the so-calle soft ) problem. Theorem 1.4. (Gromov, 1969). Suppose M is an open manifol. Then every nonegenerate 2-form on M (as long as it exists) is homotopic through nonegenerate 2-forms on M to a symplectic structure. The case where M is close turns out to be much harer. Gromov s technique (the so-calle h-principle ) oes not work in this case, but for a long time no one has been able to provie a nagetive example until C. Taubes work on the Seiberg-Witten invariants of symplectic 4-manifols. This work reveals an important connection between the symplectic topology an ifferential topology of 4-manifols.

3 LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 3 Theorem 1.5. (Taubes, 1994). The 4-manifol M = CP 2 #CP 2 #CP 2 is almost complex an has a cohomology class a H 2 (M) such that a 2 0, however, M oes not amit any symplectic structure. Now we give some natural examples of symplectic manifols. Example 1.6. (Eucliean Spaces). The most basic examples are the Eucliean spaces R 2n equippe with the stanar symplectic structure ω 0 = x 1 y 1 + x 2 y x n y n. Show that with R 2n C n uner z j = x j + iy j, j = 1, 2, n, ω 0 = i n z j z j. 2 j=1 Example 1.7. (Cotangent Bunles). Let L be a smooth manifol of imension n an M T L be the cotangent bunle. There is a canonical symplectic structure on M which has the form ω = λ. The 1-form λ is efine 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 T v M. With this unerstoo, λ is efine 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 q 1,, q n be local coorinates on L, an p 1,, p n be the corresponing coorinates on the fibers, i.e, if a cotangent vector v = n j=1 p jq j, then v has coorinates p 1,, p n on the fiber. Together q 1,, q n an p 1,, p n form a system of local coorinates on M T L. In the above local coorinates, we claim λ = n j=1 p jq j. Accepting it momentarily, we see immeiately that ω λ is a symplectic structure, because ω = n j=1 q j p j in these coorinates. To see the claim λ = n j=1 p jq j, recall that the value of λ at v = n j=1 p jq j equals π (v) = n j=1 p jπ (q j ) = n j=1 p j(π q j ). Since q 1,, q n are regare as local coorinates on M T L, q j = π q j, an with this unerstoo, v = n j=1 p jq j has coorinates q 1,, q n, p 1,, p n. This shows that λ = n j=1 p jq j. It is interesting to check that λ is characterize by the following property: for any 1-form σ on L, which can be regare as a section of the cotangent bunle π : M T L L, one has σ λ = σ. We remark that cotangent bunles form a funamental class of symplectic manifols. These are the phase spaces in classical mechanics, with coorinates q an p corresponing to position an momentum. Example 1.8. (Orientable Surfaces). Let M be a 2-imensional orientable manifol, an let ω be any volume form on M. Then (M, ω) is a symplectic manifol because in this case ω is automatically nonegenerate an close. Example 1.9. (Kähler Manifols). Let M be a complex manifol of imension n, an let h be a Hermitian metric on M. Then in a local complex coorinates z 1, z 2,, z n, h may be written as h = n j,k=1 h j k z j z k, where (h j k) is a n n Hermitian matrix, i.e., h j k = h k j. The associate 2-form to h is ω = i n j,k=1 h j k z j z k. With

4 4 WEIMIN CHEN, UMASS, SPRING 07 this unerstoo, h is calle a Kähler metric if ω is close. ω is clearly nonegenerate, hence the unerlying real manifol, also enote by M, is a 2n-imensional symplectic manifol with a symplectic structure ω. (Note that the nonegeneracy conition on ω follows from the ientity ω n = i n et(h j k) z 1 z 1 z n z n. Verify the ientity.) An important aspect of symplectic geometry is its connection with almost Kähler geometry. One can regar M as a 2n-imensional real manifol equippe with an (integrable) almost complex structure J. In this context h can be regare as a J-invariant complex symmetric bilinear form on the complexification of the almost complex vector bunle T M, an ω an h are relate by h(, ) = ω(, J( )). The complex projective spaces CP n form a funamental class of Kähler manifols. There is a canonical Kähler metric on CP n, calle the Fubini-Stuy metric. In terms of the homogeneous coorinates z 0, z 1,, z n of CP n, the associate Kähler form of the Fubini-Stuy metric is ω 0 = i 2π ( n κ=0 z κ z κ ) 2 n ( z j z j z k z k z j z k z j z k ). k=0 j k (We point out that ω 0 is normalize such that CP ω n n 0 = 1.) Note that a complex submanifol of CP n is naturally a Kähler manifol with the inuce metric. Example (Prouct of Symplectic Manifols). Given two symplectic manifols (M 1, ω 1 ), (M 2, ω 2 ), the prouct M 1 M 2 is also symplectic with symplectic structures π1 ω 1 π2 (±ω 2). Here π i : M 1 M 2 M i, i = 1, 2, an we canonically ientity T (M 1 M 2 ) with π1 (T M 1 ) π2 (T M 2 ). A symplectomorphism of a symplectic manifol (M, ω) is a iffeomorphism ψ Diff(M) which preserves the symplectic structure ω = ψ ω. Note that a symplectomorphism is particularly a volume-preserving iffeomorphism (hence is necessarily an orientation-preserving iffeomorphism), as one has ψ (ω n ) = (ψ ω) n = ω n. There is an abunance of symplectomorphisms on a symplectic manifol. In fact, the group of symplectomorphisms, enote by Symp(M, ω) or simply by Symp(M), is an infinite imensional Lie group, in contrast with the finite imensionality of the isometry group of a Riemannian metric. It has been an open question as whether symplectomorphisms iffer from volumepreserving iffeomorphisms, until in 1985 M. Gromov publishe the revolutionary paper on pseuoholomorphic curves in symplectic manifols, where he prove the following celebrate non-squeezing result. Theorem (Gromov, 1985). One can not squeeze the unit ball B 2n (1) R 2n into a long an thin cyliner B 2 (r) R 2n 2 R 2n (where r < 1) via an embeing which preserves the symplectic structure ω 0 on R 2n, however, one can always squeeze the ball into the long an thin cyliner if only the volume form ω0 n is require to be preserve.

5 LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 5 The nonegeneracy conition of a symplectic structure ω gives rise to the following canonical isomorphism T M T M : X ι(x)ω = ω(x, ). A vector fiel X is calle symplectic if ι(x)ω is close. The next result is one consequence of the closeness of a symplectic structure, which shows that when M is close, the set of symplectic vector fiels form the Lie algebra of the group Symp(M). Proposition Let M be a close manifol. If t ψ t Diff(M) is a smooth family of iffeomorphisms generate by a family of vector fiels X t via t ψ t = X t ψ t, ψ 0 = i, then ψ t Symp(M) for every t iff X t is a symplectic vector fiel for every t. Proof. Recall Cartan s formula for the Lie erivative L X ω = ι(x)ω + (ι(x)ω). Now the closeness of ω, i.e., ω = 0, implies that t ψ t ω = ψ t (L Xt ω) = ψ t ((ι(x t )ω), which vanishes if an only if ι(x t )ω is close. This proves that ψ t Symp(M) for every t iff X t is a symplectic vector fiel for every t. There is a simple way to obtain symplectic vector fiels, an therefore to obtain symplectomorphisms, which shows their abunance. Let H be a smooth function (which has a compact support if M is not close). There is a vector fiel X H canonically associate to H by the following equation The flow ψ t H on M generate by X H, i.e., ι(x H )ω = H. t ψt H = X H ψ t H, is calle a Hamiltonian flow. Note that X H is a symplectic vector fiel as ι(x H ), being exact, is close. Thus ψh t is a symplectomorphism for each t. The function H is calle the Hamiltonian function an the vector fiel X H is calle the Hamiltonian vector fiel. More generally, a symplectomorphism ψ Symp(M) is calle Hamiltonian if there exists a smooth family of ψ t Symp(M), t [0, 1], with ψ 0 = i, ψ 1 = ψ, such that the corresponing (time-epenent) symplectic vector fiel X t generating ψ t is Hamiltonian, i.e., the 1-form ι(x t )ω is exact for each t an has the form ι(x t )ω = H t for a time-epenent smooth function H t on M. The function H t is calle a time-epenent Hamiltonian function, an ψ t is calle an Hamiltonian isotopy.

6 6 WEIMIN CHEN, UMASS, SPRING 07 Example (A Hamiltonian S 1 -Action). Consier the symplectic manifol (M, ω) where M is the unit sphere in R 3, i.e. M = {(x 1, x 2, x 3 ) x x x 2 3 = 1}, an ω is the area form inuce from the embeing M R 3. We will show that the Hamiltonian flow on M associate to the height function H(x 1, x 2, x 3 ) x 3 is the S 1 -action on M efine by the rotation about the x 3 -axis. To see this, we change x 1, x 2 into polar coorinates r, θ, an note that in terms of r, θ, x 3 the volume form of R 3 is rr θ x 3. On the unit sphere M, r = 1 so that the restriction of the volume form on M is v = r θ x 3. This implies that ω = ι( r )v = θ x 3. Recall that the Hamiltonian vector fiel X H is efine by ι(x H )ω = H. It follows that for H = x 3, X H = θ, an therefore the associate Hamiltonian flow is given by the rotation about the x 3 -axis. The following question is a basic one concerning Hamiltonian S 1 -actions. When im M = 2 or 4, the question is completely unerstoo, an for the case of im M = 6, partial results have been obtaine. Question If a symplectic S 1 -action has a fixe point, is it necessarily Hamiltonian? Is there a characterization of Hamiltonian S 1 -actions in terms of fixe points ata? Now we go back to the iscussion of examples of symplectic manifols. Note that the examples we gave earlier of close symplectic manifols are all coming from Kähler manifols. (The example of orientable surfaces is Kähler because every almost complex structure in imension 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 (Symplectic > Kähler). How much larger is the category of symplectic manifols than the category of Kähler manifols? Consier the following way of constructing symplectic manifols. Suppose (M, ω) is a symplectic manifol an a iscrete group G acts on M via symplectomorphisms, i.e., for every g G, the iffeomorphism g : M M is an element of Symp(M, ω). Suppose furthermore that the action of G is free, an hence the quotient M/G is again a smooth manifol. Then ω escens to a symplectic structure on the quotient manifol M/G, with which M/G naturally becomes a symplectic manifol. For a typical example, consier R 2n with the stanar symplectic structure ω 0 = n j=1 x j y j, an the free action by translation of G = Z 2n the integral lattice. The quotient is the (2n)- imensional torus T 2n, with a stanar symplectic structure. The following example is also of this sort, which gives the first example of a close, symplectic but non-kähler manifol. (The example was known to Koaira in the 1950s an was reiscovere in the 1970s by Thurston.)

7 LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 7 Example (A Non-Kähler Manifol). Consier the group G = Z 2 Z 2 with the noncommutative group operation ( ) (j, k ) (j, k) = (j + j, A j k + k 1 j2 ), A j =, 0 1 where j = (j 1, j 2 ) Z 2, an similarly for k. This group acts on R 4 via G Diff(R 4 ) : (j, k) ρ jk, where ρ jk (x, y) = (x + j, A j y + k). One can verify easily that the action is free an preserves the symplectic structure ω = x 1 x 2 + y 1 y 2. Hence the quotient M = R 4 /G is a symplectic manifol which is easily seen to be close. We will show that H 1 (M; Z) = Z 3 so that M is not a Kähler manifol. (Recall that by the Hoge theory the o imensional Betti numbers of a close Kähler manifol must be even.) To see this, one first verifies that the commutator subgroup [G, G], which consists of elements of the form aba 1 b 1, a, b G, equals 0 0 Z 0. Then note that π 1 (M) = G an H 1 (M; Z) = π 1 (M)/[π 1 (M), π 1 (M)]. Hence H 1 (M; Z) = G/[G, G] = Z 3. This implies that the first Betti number of M equals 3, an that M is not Kähler. Topologically, M is a nontrivial T 2 bunle over T 2, or more precisely, M = S 1 N where N is the nontrivial T 2 bunle over S 1 efine as follows. Let x, y be the stanar coorinates on T 2. Then N = [0, 1] T 2 /, where (0, x, y) (1, x + y, y). 2. Submanifols of Symplectic Manifols Let (M, ω) be a symplectic manifol, an let Q M be a submanifol of M. Note that the tangent bunle T Q is a sub-bunle of T M Q. We set T Q ω {u T M q ω(u, v) = 0 v T Q q }, q Q which is also a sub-bunle of T M Q. We say Q is isotropic if T Q T Q ω, coisotropic if T Q ω T Q, symplectic if T Q T Q ω = {0}, an Lagrangian if T Q = T Q ω. Amongst these four types of submanifols of a symplectic manifol, Lagrangian submanifols form the most important class, which is what we shall concentrate in this section. Note that a Lagrangian submanifol in a 2n-imensional symplectic manifol has imension n. Example 2.1. (Totally real submanifols in a Kähler manifol). Let M be a Kähler manifol of complex imension n. A real n-imensional submanifol Q is calle totally real if for every point q Q there is a local holomorphic complex coorinates centere at q within which Q is ientifie with the real part R n C n. Check that a totally real submanifol in a Kähler manifol is Lagrangian with respect to the Kähler form. When n = 1, every symplectic manifol is Kähler, an every real 1-imensional submanifol is Lagrangian an totally real. For higher imensional examples, suppose M is a complex submanifol of CP N, which is Kähler uner the inuce metric. Then the

8 8 WEIMIN CHEN, UMASS, SPRING 07 fixe-point set of the anti-holomorphic involution z i z i in M, if nonempty, is a totally real submanifol of M. Example 2.2. (Lagrangian submanifols in (R 2n, ω 0 )). The n-imensional spaces efine by x j = constant, j = 1,, n, or y j = constant, j = 1,, n, are Lagrangian submanifols in (R 2n, ω 0 ). For a ifferent type of examples, consier the n-torus T n R 2n, where we think (R 2n, ω 0 ) as a prouct of symplectic manifol (R 2, ω 0 ) an T n S 1 S 1, where the j-th copy of S 1 is the unit circle {x 2 j + y2 j = 1} in the j-th copy of R2. The torus T n is Lagrangian because (1) any 1-imensional manifol in a 2-imensional symplectic manifol is Lagrangian, an (2) the prouct of Lagrangian submanifols in a prouct symplectic manifol is again Lagrangian. Example 2.3. (Lagrangian submanifols an symplectomorphisms). Let (M, ω) be any symplectic manifol. Then in the prouct symplectic manifol (M M, ω ( ω)), the iagonal {(x, x) M M x M} is Lagrangian. More generally, for any ψ Symp(M, ω), the gragh of ψ in M M, is a Lagrangian submanifol. gragh(ψ) {(x, ψ(x)) M M x M}, Example 2.4. (Lagrangian submanifols in cotangent bunles). Let M T L be the cotangent bunle of L equippe with the canonical symplectic structure ω = λ. (In local coorinates λ = j p jq j an ω = j q j p j.) Clearly, the fibers of T L (efine by q j = constant, j) an the zero section L T L (efine by p j = 0, j) are Lagrangian submanifols. Next we consier submanifols Q σ in M which is the graph of a 1-form σ on L, regare as a smooth section of T L. Since Q σ is of half imension of M, it follows that Q σ is a Lagrangian submanifol if an only if the pull-back of ω to Q σ equals 0, which is equivalent to the conition that the pull-back σ ω = 0. But σ ω = σ ( λ) = (σ λ) = σ, which implies that Q σ is Lagrangian iff σ is close. One of the funamental questions in symplectic topology concerns the intersection of Lagrangian submanifols. In orer to motivate this question, we consier the intersection of Q σ for a nonzero close 1-form σ with the zero section Q 0 = L. Observe that the set of intersection points of Q σ an L is precisely the set of zeroes of σ on L. With this unerstoo, one shoul istinguish the following two cases: (1) σ = f is exact, (2) σ is not exact. In the first case, Q σ is calle an exact Lagrangian submanifol, an the function f : L R is calle a generating function of Q σ. Now the basic observation is that on a close manifol exact 1-forms σ = f have more zeroes (which correspon to the critical points of the function f) than close 1-forms in general. For instance, on the 2-torus T 2, any smooth function has at least 3 critical points an any Morse function has at least 4 critical points (which equals the sum of the Betti numbers), while there are close 1-forms on T 2 which are nowhere vanishing.

9 LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 9 Lemma 2.5. Suppose σ is a close 1-form on L an Q σ is the corresponing Lagrangian submanifol in T L. Let ψ t Symp(T L), t [0, 1], be an isotopy of symplectomorphisms such that ψ 0 = i an ψ 1 (L) = Q σ. Then σ is exact if ψ t is an Hamiltonian isotopy. Proof. Let X t be the corresponing time-epenent symplectic vector fiel. Then t ψ t λ = ψ t (L Xt λ) = ψ t (ι(x t )λ + (ι(x t )λ)), which implies that ψ t λ λ is exact for each t if ψ t is an Hamiltonian isotopy, i.e., ι(x t )ω = ι(x t )λ is exact for every t. The lemma follows by observing that, since ψ 1 (L) = Q σ, σ = σ λ = i (ψ 1λ λ), where i : L T L is the zero section. Problem 2.6. (Arnol s conjecture on Lagrangian intersections). Let L 0, L 1 be compact Lagrangian submanifols of a symplectic manifol (M, ω). Moreover, suppose that ψ t Symp(M, ω), t [0, 1], is an Hamiltonian isotopy such that ψ 0 = i an L 1 = ψ 1 (L 0 ). Then L 0 an L 1 must have at least as many intersection points as a function on L 0 must have critical points. Let (M, ω) be a close symplectic manifol, an let ψ Symp(M, ω). Then as we have seen earlier that both the iagonal an the graph of ψ are Lagrangian submanifols of the prouct (M M, ω ( ω)). Moreover, observe that the intersection points of an the graph of ψ are exactly the fixe points of ψ in M, i.e., x M such that ψ(x) = x. Now suppose ψ t is an Hamiltonian isotopy on M. Then Ψ t (i, ψ t ) is an Hamiltonian isotopy on M M, an Ψ t ( ) is the graph of ψ t for every t. Problem 2.7. (Arnol s conjecture on symplectic fixe points). Let ψ : M M be an Hamiltonian symplectomorphism of a close symplectic manifol (M, ω). Then ψ must have at least as many fixe points as a function on M must have critical points. Note that Arnol s conjecture on symplectic fixe points is trivially true when the Hamiltonian symplectomorphism ψ is generate by a time-inepenent Hamiltonian H on M. Inee, in this case, the symplectic fixe points are precisely the critical points of the Hamiltonian function H. We remark that Arnol s conjectures have been one of the focal points of much of the recent evelopement in symplectic topology. The celebrate Floer Homology theory was initially introuce to tackle these conjectures! 3. Contact Manifols Contact topology is the o-imensional analogue of symplectic topology. While contact topology is an interesting subject of its own right, the interplay between symplectic topology an contact topology has become a tren in recent years. Definition 3.1. Let N be a (2n + 1)-imensional manifol, an let ξ T N be a hyperplane fiel, which is efine by a 1-form α, i.e., ξ = ker α. ξ is calle a contact

10 10 WEIMIN CHEN, UMASS, SPRING 07 structure if α is nonegenerate when restricte to ξ, or equivalently, α (α) n 0. The 1-form α is calle a contact form associate to ξ, an the pair (N, ξ) is calle a contact manifol. Remark 3.2. (1) Recall that the Frobenius integrability theorem says that a hyperplane fiel ξ = ker α is integrable if an only if α α = 0. So a contact structure may be thought of as a maximally nonintegrable istribution. (2) Note that α (α) n is a volume form on N, in particular, N is orientable, an is canonically oriente by α (α) n. On the other han, α is nonegenerate on ξ, so that ξ is canonically oriente by (α) n. These together etermines an orientation of the quotient bunle T N/ξ. In other wors, the contact structure ξ is both oriente an co-oriente. (3) Suppose α is another 1-form which efines ξ, i.e., ξ = ker α. Then α = fα where f is a nowhere vanishing function. One can easily check that α (α ) n = f n+1 α (α) n 0. Hence α is another contact form associate to ξ, an ξ being a contact structure is inepenent of the choice of its efining 1-form. We emphasize that the contact structure is the hyperplane fiel ξ not the contact form α. It etermines the contact forms up to a nonzero factor f. We usually assume that f > 0 so that the relevant orientations of the contact structure efine through the contact forms are compatible. Definition 3.3. Let (N, ξ) be a contact manifol of imension 2n + 1, an let L be a n-imensional submanifol of N. L is calle Legenrian if T L ξ L. In a certain sense Legenrian submanifols are contact analogues of Lagrangian submanifols. Suppose α is any contact form associate to a contact structure ξ, an suppose L is a Legenrian submanifol. Then α = 0 on T L. To see this, let X, Y be any vector fiels on L. Then [X, Y ] is also a vector fiel on L. Since T L ξ L, we have α(x) = α(y ) = α([x, Y ]) = 0, which implies that α(x, Y ) = 0. Example 3.4. (Eucliean spaces). Let x 1,, x n, y 1,, y n, z be the coorinates on R 2n+1. Then the 1-form n α 0 = z y j x j is a contact form on R 2n+1. Let s visualize the corresponing contact structure ξ 0 for the case of R 3. Let x, y, z be the coorinates on R 3. Then α 0 = z yx. It is easily seen that ξ 0 = {a y + b( x + y z ) a, b R}. (x,y,z) R 3 Example 3.5. (1-jet bunles). Let L be a n-imensional manifol, an let N T L R be the 1-jet bunle over L. Then j=1 α = z λ

11 LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 11 is a contact form on N, where z is the coorinate on the R factor an λ is the canonical 1-form on the T L factor. Note that for any smooth function f : L R, the submanifol L f {(x, f(x), f(x)) x L} N is Legenrian. Example 3.6. (Circle bunles). Let N be a circle bunle over Σ. Then for any connection 1-form A Ω 1 (N; ir) of the bunle, F A = A escents to a 2-form on Σ such that the first Chern class of the circle bunle π : N Σ is represente by if A /2π. Now suppose that the first Chern class of the circle bunle π : N Σ is represente by a 2-form ω which is a symplectic structure on Σ. Then one can choose a connection 1-form A such that F A = 2πiπ ω. We set α 1 2π A. Then α is a contact 1-form on N with α = π ω. One basic example of the above construction is given by the Hopf fibration π : S 2n+1 CP n, where the first Chern class is represente by the Kähler form of the Fubini-Stuy metric on CP n. Let (N, ξ) be a contact manifol an α be a contact form associate to the contact structure ξ. A iffeomorphism ψ Diff(N) is calle a contactomorphism if ψ preserves the oriente hyperplane fiel ξ, or equivalently, if there is a smooth function h such that ψ α = e h α. A contact isotopy is a smooth family ψ t of contactomorphisms such that ψt α = e ht α for a smooth family of functions h t. A vector fiel X on a contact manifol (N, ξ) is calle a contact vector fiel if L X α = gα for a contact form α associate to ξ, where g is a smooth function on N. Consier a time-epenent contact vector fiel X t, such that L Xt α = g t α, an the smooth family of iffeomorphisms ψ t generate by X t, i.e., t ψ t = X t ψ t, ψ 0 = i. One can check easily that ψ t is a contact isotopy. In fact, since t ψ t α = ψt L Xt α = ψt g t α, one has ψt α = e ht α where h t = t 0 ψ sg s s. Conversely, if ψ t is a contact isotopy with ψt α = e ht α, then L Xt α = g t α where g t = (ψt 1 ) t h t. Given any contact form α, there is a canonical contact vector fiel X α, calle a Reeb vector fiel, which has the property L Xα α = 0. X α is uniquely etermine by the conitions ι(x α )α = 0 an ι(x α )α = 1. The ynamical system generate by a Reeb vector fiel is calle a Reeb ynamics. More generally, fixing a contact form α, there is a 1-1 corresponence between contact vector fiels an smooth functions as follows: ι(x H )α = H an ι(x H )α = H (ι(x α )H)α, where H is a smooth function an X H is the corresponing contact vector fiel. Note that X α correspons to H 1.

12 12 WEIMIN CHEN, UMASS, SPRING 07 We close this section with a iscussion on the interaction between contact topology an symplectic topology. Recall that symplectic geometry is the geometry on which Hamiltonian mechanics rests. One of the funamental problems in Hamiltonian mechanics concerns the existence of perioic solutions, i.e., perioic orbits of the corresponing Hamiltonian ynamical system. More concretely, suppose (M, ω) is a symplectic manifol an H is a Hamiltonian function on M such that each level surface H 1 (c) is compact. Let X H be the corresponing Hamiltonian vector fiel, efine by ι(x H )ω = H. Note that the Hamiltonian ynamical system generate by X H preserves the level surfaces of H as seen easily from the fact X H (H) = ω(x H, X H ) = 0. In particular, the flow ψ t efine by t ψ t = X H ψ t, ψ 0 = i exists for t (, ) since each level surface H 1 (c) is compact. A perioic orbit of the Hamiltonian ynamical system is a map γ : R M which satisfies t γ(t) = X H(γ(t)), t R, an γ(t ) = γ(0) for some T > 0. Note that every perioic orbit is containe in some level surface of H. Given any contact manifol (N, ξ) with a contact form α, there is a canonical symplectic structure on R N efine as follows: let t be the coorinate on the R factor, then the 2-form (e t α) is a symplectic structure on R N. The symplectic manifol (R N, (e t α)) is calle the symplectization of (N, α). Note that the vector fiel t on R N has the property that L t ω = ω where ω (e t α) is the canonical symplectic structure on R N. Such a vector fiel is calle a Liouville vector fiel. Let (M, ω) be a symplectic manifol, an 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 neighborhoo of N in M can be symplectically ientifie with a neighborhho of {0} N in the symplectization of (N, α). For example, the unit sphere S 2n 1 in (R 2n, ω 0 ) is a hypersurface of contact type with contact form α 0 = 1 n (x j y j y j x j ). 2 j=1 Problem 3.7. (Weinstein s conjecture on Hamiltonian ynamics). Suppose a level surface H 1 (c) is of contact type in (M, ω). Then there exists a perioic orbit of X H containe in H 1 (c). Note that suppose H 1 (c) is of contact type an α is the contact form on H 1 (c) with respect to which a neighborhoo of H 1 (c) in (M, ω) is ientifie symplectically with a neighborhho of {0} H 1 (c) in the symplectization. Then it is easily seen that ω H 1 (c) = α. In particular, X H is in the null irection of α, i.e., ι(x H )α = 0. Notice that the perioic orbits of X H are in 1-1 corresponence with the perioic orbits of the Reeb vector fiel X α on H 1 (c).

13 LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 13 Problem 3.8. (Weinstein s conjecture on Reeb ynamics). Let N be a compact contact manifo with contact form α. Then there exists at least one perioic orbit of the Reeb vector fiel X α. For another aspect of contact topology in symplectic geometry, let s consier symplectic structures on a manifol M with bounary. Suppose M is almost complex, or equivalently, M amits a nonegenerate 2-form. Then by Gromov s theorem the interior of M amits a symplectic structure. Of course, in general one has no a priori knowlege about the behavior of the symplectic structure near the bounary M. Let M be a manifol with bounary, an let ω be a symplectic structure on M. We say (M, ω) is a symplectic manifol with contact bounary if M is of contact type an a neighborhoo of M in (M, ω) is symplectically ientifie with ( ɛ, 0] M in the symplectization for some ɛ > 0. For example, the unit ball B 2n (1) R 2n with the stanar symplectic structure ω 0 is a symplectic manifol with contact bounary. While Gromov s work shows that open manifols have a rather trivial symplectic topology, close symplectic manifols exhibit many rigiity properties, iscovere mainly through his powerful theory of pseuoholomorphic curves. Symplectic manifols with contact bounary resemble in many aspects close symplectic manifols, because the pseuoholomorphic curve theory can be properly extene to this context. Symplectic geometry also provies a useful tool in the stuy of contact topology, especially in imension 3. A compact contact manifol (N, ξ) is calle symplectically fillable if (N, ξ) can be realize as the bounary of a symplectic manifol with contact bounary. For example, the contact structure on S 3 obtaine via the Hopf fibration S 3 CP 1 as escribe in Example 3.6 is symplectically fillable; it can be realize as the bounary of the symplectic manifol with contact bounary (B 4 (1), ω 0 ). Symplectically fillable contact structures on a 3-manifol belong to a class of contact structures, calle tight contact structures, which exhibit many rigiity properties. References [1] D. McDuff an D. Salamon, Introuction to Symplectic Topology, Oxfor Mathematical Monographs, 2n eition, Oxfor Univ. Press, 1998.