LECTURE 2: SYMPLECTIC MANIFOLDS Contents 1. Symplectic Manifolds 1 2. The cotangent bundle 4 3. Coadjoint orbits 8 1. Symplectic Manifolds ¶ Definitions. Let M be a smooth manifold, and ω ∈ Ω 2 (M) = Γ∞(Λ2T ∗M) a smooth 2-form on M. Recall that by definition this means that for any p ∈ M, ωp : TpM × TpM → R is a skew-symmetric bilinear map, and ωp depends smoothly on p. Definition 1.1. We call ω a symplectic form on M if (1) (closeness) ω is a closed 2-form, i.e. dω = 0 (2) (non-degeneracy) for ∀p ∈ M, ωp is a linear symplectic form on TpM. We will call the pair (M, ω) a symplectic manifold. Remarks. Let (M, ω) be a symplectic manifold. According to the linear theory, • dim M = dim TpM must be even. • If we denote dim M = 2n, then ω n p 6= 0 ∈ Λ 2n (T ∗ p M), i.e. ω n is a non-vanishing 2n form, thus a volume form, on M. As a consequence of the existence of a volume form, we see M must be orientable. Definition 1.2. We will call ω n n! the symplectic volume form or the Liouville volume form of (M, g). • If ω is not only closed but also exact, i.e. there exists a 1-form α on M so that ω = dα, then we say (M, ω) is an exact symplectic manifold. 1
2 LECTURE 2: SYMPLECTIC MANIFOLDS ¶ Examples. Example. (R 2n , Ω0) is of course the simplest symplectic manifold. Example. Let S be any oriented surface and ω any volume form on S. Then • ω is non-degenerate since it is a volume form; • ω is closed since it is already a top form. In other words, (S, ω) is symplectic. Example. If (M, ω) is a symplectic manifold, and U ⊂ M an open subset, then (U, ω) is a symplectic manifold. Example. If (M, ω) is a symplectic manifold, then for any λ 6= 0, (M, λω) is a symplectic manifold. Example. If (M1, ω1) and (M2, ω2) are symplectic manifolds, then for any nonzero real numbers λ1, λ2, the 2-form ω = λ1ω1⊕λ2ω2 is a symplectic form on the product manifold M = M1 × M2. Example. The 2n-dimensional torus T 2n = R 2n/Z 2n , with its standard symplectic form ω0 = Pdxj ∧ dyj , is a symplectic manifold. Example. We will see more examples later, including cotangent bundles, coadjoint orbits, K¨ahler manifolds etc. (So, yes, we have “as many” symplectic manifolds as smooth manifolds! ) ¶ Distinguished submanifolds. As in the linear case we can define Definition 1.3. Let (M, ω) be a 2n-dimensional symplectic manifold and Z ⊂ M a smooth submanifold. We call Z • isotropic if for all p ∈ Z, TpZ is an isotropic subspace of (TpM, ωp). • coisotropic if for all p ∈ Z, TpZ is a coisotropic subspace of (TpM, ωp). • Lagrangian if for all p ∈ Z, TpZ is a Lagrangian subspace of (TpM, ωp). • symplectic if for all p ∈ Z, TpZ is a symplectic subspace of (TpM, ωp). The following properties follows immediately from the corresponding linear ones: (1) ι : Z ,→ M is a symplectic submanifold if and only if ι ∗ω is symplectic. (2) ι : Z ,→ M is an isotropic submanifold if and only if ι ∗ω = 0. (3) ι : Z ,→ M is a Lagrangian submanifold if and only if ι ∗ω = 0 and dim Z = 1 2 dim M. (4) If Z is an isotropic submanifold of M, then dim Z ≤ 1 2 dim M. (5) If Z is an coisotropic submanifold of M, then dim Z ≥ 1 2 dim M. (6) Any one dimensional submanifold is isotropic, any co-dimension one submanifold is coisotropic
LECTURE 2: SYMPLECTIC MANIFOLDS 3 Lagrangian submanifolds are in some sense the most important object (according to Weinstein) in symplectic manifolds. We will study more details of Lagrangian submanifolds later. ¶ More topological restrictions. As we have seen, to be a symplectic manifold, M must be even dimensional and oriented. There are more obstructions: Proposition 1.4. Let (M, ω) be a closed (=compact without boundary) symplectic manifold of dimension 2n. Then for any 1 ≤ k ≤ n, the de Rham cohomology group H2k (M) 6= {0}. Proof. Since ω is closed, [ω] ∈ H2 (M) is a de Rham cohomology class. So for any 1 ≤ k ≤ n, [ω k ] ∈ H2k (M). If H2k (M) = {0}, then ω k = dα for some α ∈ Ω 2k−1 (M). It follows ω n = ω k ∧ ω n−k = d(α ∧ ω n−k ), which is impossible since ω n is a volume form, and a volume form on a closed manifold cannot be exact. As an immediate consequence, we see that S 2 is the only sphere that admits a symplectic structure: Corollary 1.5. For any n > 1, the sphere S 2n admits no symplectic structure. Proof. In algebraic topology one can prove H2 (S 2n ) = 0 for any n > 1. What we really proved is that [ω] 6= 0 ∈ H2 (M) if M is compact. So Corollary 1.6. Any exact symplectic manifold is non-compact. Using a similar argument one gets Proposition 1.7. Any symplectic submanifold of (R 2n , ω) is non-compact, where ω is any symplectic form on R 2n . Proof. Since R 2n is contractible, any closed 2-form is exact. It follows that there exists a one-form α on M so that ω = dα. As a consequence, for any m, the 2m form ω m = d(α ∧ ω m−1 ) is exact on R 2n , and thus exact on any submanifold. On the other hand, if M2m is a compact symplectic submanifold of R 2n , then ω m is a volume form on M, which is a contradiction since on a compact manifold the volume form can never be exact. Remark. On the other hand, One can easily construct many symplectic submanifolds. For example, one can identify the symplectic manifold R 2n with the complex manifold C n . Then any complex submanifold is a symplectic submanifold. Remark. There exists more subtle obstructions for the existence of symplectic structure. For example, the existence of a symplectic structure implies the existence of an almost complex structure, which gives an obstruction in the Pontryagin class.
4 LECTURE 2: SYMPLECTIC MANIFOLDS ¶ Symplectomorphisms. As in the linear case, we can define Definition 1.8. Let (M1, ω1) and (M2, ω2) be symplectic manifolds. A smooth map f : M1 → M2 is called a symplectomorphism (or a canonical transformation) if it is a diffeomorphism and (1) f ∗ω2 = ω1. We say (M1, ω1) is symplectomorphic to (M2, ω2) if such a symplectomorphism exists. Next time we will prove : Theorem 1.9 (Darboux theorem). Let (M, ω) be a symplectic manifold of dimension 2n. Then for any p ∈ M, there exists a neighborhood U of p in M and a neighborhood U0 of 0 in R 2n so that (U, ω) is symplectomorphic to (U0, Ω0), where Ω0 is the standard (linear) symplectic form on R 2n . Remark. (1) In the language of local coordinates, this means that near any p ∈ M, one can find coordinate patch (U, x1, · · · , xn, ξ1, · · · , ξn) centered at p such that on U, ω = Xdxi ∧ dξi . Definition 1.10. This coordinate patch is called a Darboux coordinate patch. (2) As a consequence we see that for symplectic manifolds there is no local geometry: locally all symplectic manifolds of the same dimension look the same. So symplectic geometry is very different from Riemannian geometry where one has many local geometric quantities (like curvature). (However, there are much to say about the global geometry/topology of symplectic manifolds!) 2. The cotangent bundle ¶ The canonical symplectic structure on cotangent bundles. Let X be an n-dimensional smooth manifold and denote by M = T ∗X its cotangent bundle. Let π : T ∗X → X, (x, ξ) 7→ x be the bundle projection map. In manifold theory we learned that T ∗X is a smooth manifold of dimension 2n, and any local coordinate system on X induces an coordinate system in T ∗X, as follows: From any coordinate patch (U, x1, · · · , xn) of X one can construct, for x in U, a basis {∂1, · · · , ∂n} of TxX, which depends smoothly in x. We denote the dual basis of T ∗ xX by {dx1, · · · , dxn}. This gives the induced coordinate system, (x1, · · · , xn, ξ1, · · · , ξn), on MU = π −1 (U). Namely, if ξ ∈ T ∗ xX, then ξ = Xξi(dxi)x
LECTURE 2: SYMPLECTIC MANIFOLDS 5 For any p = (x, ξ) ∈ M, we let (2) αp = (dπp) ∗ ξ. Note that by definition ξ ∈ T ∗ xX, so for any p ∈ T ∗X, αp = (dπp) ∗ ξ ∈ T ∗ p (T ∗X). In other words, we get a smooth 1-form α ∈ Ω 1 (M) = Γ∞(T ∗ (T ∗X)). Note that for any vp ∈ TpM, hαp, vpip = h(dπp) ∗ ξ, vpip = hξ, dπp(vp)ix. Definition 2.1. We call α the canonical 1-form (or tautological 1-form) on T ∗X. Proposition 2.2. In local coordinates described above, (3) α = Xn i=1 ξidxi . Proof. Let vp = Pn i=1(ai ∂ ∂xi + bi ∂ ∂ξi ) ∈ TpM be an arbitrary tangent vector. Then hαp, vpip = hξ,(dπp)vpix = h Xξi(dxi)x, Xai ∂ ∂xi ix = Xaiξi = h Xξidxi , vpip. The equation follows. As a consequence, if we let (4) ω = −dα, then ω is closed, and from the local expression of α, (5) ω = Xdxi ∧ dξi we see that ωp is also non-degenerate for all p. It follows that ω is a symplectic form on M. Definition 2.3. We call ω = −dα the canonical symplectic form on M = T ∗X. Remark. In the future we will denote the canonical 1-form αcan, and the canonical symplectic form ω by ωcan. Obviously (T ∗X, ωcan) is an exact symplectic manifold, and any coordinate patch on X induces a Darboux coordinate patch on (T ∗X, ωcan).
6 LECTURE 2: SYMPLECTIC MANIFOLDS ¶ The reproducing property. Let µ ∈ Ω 1 (X) be an arbitrary smooth 1-form. By definition we can identify µ with a smooth section sµ : X → M = T ∗X via sµ(x) = (x, µx). Let β ∈ Ω 1 (M) be a 1-form on M, then s ∗ µβ is a 1-form on X. It is not hard to check by definition that (s ∗ µβ)x = (dsµ) ∗ xβp. A crucial property for the canonical 1-form α ∈ Ω 1 (M) is the following Theorem 2.4 (Reproducing property). For any 1-form µ ∈ Ω 1 (X), we have (6) s ∗ µα = µ. Conversely, if α ∈ Ω 1 (M) is a 1-form such that (6) hold for all 1-form µ ∈ Ω 1 (X), then α is the canonical 1-form. Proof. At any point p = (x, ξ) we have αp = (dπp) ∗ ξ. So at p = sµ(x) = (x, µx) we have αp = (dπp) ∗µx. It follows (s ∗ µα)x = (dsµ) ∗ xαp = (dsµ) ∗ x (dπp) ∗µx = (d(π ◦ sµ))∗ xµx = µx. Conversely, suppose α0 ∈ Ω 1 (M) is another 1-form on M satisfying the reproducing property above, then for any 1-form µ ∈ Ω 1 (X), we have s ∗ µ (α −α0) = 0. So for any v ∈ TxX, 0 = h(dsµ) ∗ x (α − α0)p, vi = h(α − α0)p,(dsµ)x(v)i. For each p = (x, ξ), the set of all vectors of the this form, {(dsµ)xv | µ ∈ Ω 1 (X), µx = ξ, v ∈ TxX, } span TpM, so we conclude that α = α0. ¶ Naturality. The construction of the canonical symplectic form on cotangent bundles is natural in the following sense: Suppose X and Y are smooth manifolds of dimension n and f : X → Y a diffeomorphism. We can “lift” f to a smooth map ˜f : T ∗X → T ∗Y by (7) ˜f(x, ξ) = (f(x),(df ∗ x ) −1 (ξ)). Theorem 2.5 (Naturality). The map ˜f is a diffeomorphism and preserves the canonical 1-forms: ˜f ∗αT ∗Y = αT ∗X.
LECTURE 2: SYMPLECTIC MANIFOLDS 7 Proof. It is not hard to check that ˜f is a diffeomorphism. Denote the projections by π1 : T ∗X → X and π2 : T ∗Y → Y . By definition π2 ◦ ˜f = f ◦ π1. So if we denote ˜f(x, ξ) = (y, η), then ˜f ∗αT ∗Y = d ˜f ∗ ◦ (dπ∗ 2 )η = (dπ∗ 1 ◦ df ∗ )η = (dπ∗ 1 )ξ = αT ∗X. As a consequence, we get immediately Theorem 2.6. The map ˜f : T ∗X → T ∗Y constructed above is a symplectomorphism with respect to the canonical symplectic forms. ¶ Lagrangian submanifolds of T ∗X. By definition, a Lagrangian submanifold Λ of (T ∗X, ωcan) is a smooth submanifold of dimension dim Λ = dim X and satisfies ι ∗ωcan = 0, where ι : Λ ,→ T ∗X is the inclusion map. From the local expression of ω, it is easy to see Example. The zero section of T ∗X, X0 = {(x, ξ) ∈ T ∗X | x ∈ X, ξ = 0 ∈ T ∗ xX} is a Lagrangian submanifold of T ∗X. Example. For each x ∈ X, the cotangent fiber T ∗ xX = {(x, ξ) ∈ T ∗X | ξ ∈ T ∗ xX} is a (vertical) Lagrangian submanifold of T ∗X. The first example is the graph of the zero 1-form on X. Suppose µ ∈ Ω 1 (M) is any smooth 1-form on X. Then its graph, Xµ = {(x, µx) | x ∈ X}, is a n dimensional submanifold of T ∗X. [Such a submanifold is called horizontal since π ◦ ι is a diffeomorphism from the submanifold onto (its image in) the base manifold X.] A natural question is: when will Xµ a Lagrangian submanifolds? Proposition 2.7. Xµ is Lagrangian if and only if dµ = 0. Proof. Let ι : Xµ ,→ T ∗X be the inclusion map. Note that π ◦ ι : Xµ → X is a diffeomorphism. Let γ : X → Xµ be its inverse. Then by definition sµ = ι ◦ γ. So from the reproducing property, Xµ is Lagrangian ⇔ ι ∗ dα = 0 ⇔ γ ∗ ι ∗ dα = 0 ⇔ d(s ∗ µα) = 0 ⇔ dµ = 0.
8 LECTURE 2: SYMPLECTIC MANIFOLDS The proposition characterize the horizontal Lagrangian submanifolds of T ∗X. Of course there are many non-horizontal Lagrangian submanifolds of T ∗X, the simplest one being the vertical fiber we described just now. More generally, for any submanifold S of X, we define its conormal bundle in T ∗X to be N ∗S = {(x, ξ) ∈ T ∗X | x ∈ S, ξ ∈ T ∗ xX so that ξ(v) = 0 for all v ∈ TxS}. For example, the zero section is the conormal bundle of X itself, while the vertical fiber T ∗ xX is the conormal bundle of the zero dimensional submanifold {x}. Obviously if the dimension of S is k, then at any x ∈ S, the fiber has dimension n − k. So N∗S has dimension n, half the dimension of T ∗X. Proposition 2.8. For any smooth submanifold S of X, the conormal bundle N∗S is a Lagrangian submanifold. Proof. Suppose S is described in local coordinates by xk+1 = · · · = xn = 0, then at each x ∈ S, the conormal fiber N∗ xS is described by ξ1 = · · · = ξk = 0. It follows that the symplectic form ω = Pdxi ∧ dξi vanishes on N∗ xS. 3. Coadjoint orbits Student presentation: Qin Yuguo More details will be added.