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Lecture 1: Basic Concepts, Problems, and Examples 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 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 beyond the scope of this course or will be treated more systematically in later lectures. 1. Symplectic Manifolds Throughout we will assume that M is a C1-smooth manifold 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 ! 2 Ω2(M), which is (1) nondegenerate, and (2) closed (i.e. d! = 0). (Recall that a 2-form ! 2 Ω2(M) is said to be nondegenerate if for every point p 2 M, !(u; v) = 0 for all u 2 TpM implies v 2 TpM equals 0.) The pair (M; !) is called a symplectic manifold. Before we discuss examples of symplectic manifolds, we shall first derive some im- mediate consequences of a symplectic structure. (1) The nondegeneracy condition on ! is equivalent to the condition that M has an even dimension 2n and the top wedge product !n ≡ ! ^ ! · · · ^ ! is nowhere vanishing on M, i.e., !n is a volume form. In particular, M must be orientable, and is canonically oriented by !n. 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 !. (2) The closedness of ! is a very important, nontrivial geometric or analytical condition. For now, let us simply observe that ! defines a deRham cohomology class [!] 2 H2(M), which must be nonzero when M is closed. In fact in this case, [!]n ≡ [!] [ [!] ···[ [!] 2 H2n(M) n R n must be nonzero, since [!] ([M]) = M ! 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 2 S ⊂ R , let ν(x) be the unit outward normal vector of S at x. We define 6 Jxu = ν(x) × u; 8u 2 TxS : 6 2 2 Then Jxu 2 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 complex manifold; 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 almost complex manifold of dimension 2n where n ≥ 2, and moreover, when M is closed, there exists a cohomology class a 2 H2(M) such that 0 6= an 2 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 Riemann surface, 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 2 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 !0 = dx1 ^ dy1 + dx2 ^ dy2 + ··· + dxn ^ dyn: 2n n Show that with R ≡ C under zj = xj + iyj, j = 1; 2 ··· ; n, n i X ! = 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 cotangent bundle. There is a canonical symplectic structure on M which has the form ! = −dλ. The 1-form λ is defined as follows. Let π : M ≡ T ∗L ! L be the natural projection. ∗ ∗ ∗ ∗ Then for any v 2 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 2 Tπ(v)L, so that ∗ ∗ λ(v) ≡ π (v) 2 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 ! ≡ −dλ is a symplectic structure, Pn because ! = 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 section 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 ! be any volume form on M. Then (M; !) is a symplectic manifold because in this case ! 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 ! = i j;k=1 hjk¯dzj ^ dz¯k. With 4 WEIMIN CHEN, UMASS, SPRING 07 this understood, h is called a K¨ahlermetric if ! is closed. ! is clearly nondegenerate, hence the underlying real manifold, also denoted by M, is a 2n-dimensional symplectic manifold with a symplectic structure !. (Note that the nondegeneracy condition on ! follows from the identity n n ! = i det(hjk¯) dz1 ^ dz¯1 ^ · · · ^ dzn ^ dz¯n: Verify the identity.) An important aspect of symplectic geometry 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 ! and h are related by h(·; ·) = !(·;J(·)). n The complex projective spaces CP form a fundamental class of K¨ahlermanifolds.
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