LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND 1. Symplectic structure on cotangent bundle ¶ Linear symplectic structure. Definition 1.1. A symplectic vector space is a pair (V, Ω), where V is a real vector space, and Ω : V × V → R a non-degenerate linear 2-form.1 Ω is called a linear symplectic structure or a linear symplectic form on V . Example. Let V = R 2n = R n × R n and define Ω0((x, ξ),(y, η)) := hx, ηi − hξ, yi, then (V, Ω0) is a symplectic vector space. Let {e1, · · · , en, f1, · · · , fn} be the standard basis of R n × R n , and {e ∗ 1 , · · · , e∗ n , f ∗ 1 , · · · , f ∗ n} the dual basis of (R n ) ∗ × (R n ) ∗ , then as a linear 2-form one has Ω0 = Xn i=1 e ∗ i ∧ f ∗ i . ¶ Linear Darboux theorem. Definition 1.2. Let (V1, Ω1) and (V2, Ω2) be symplectic vector spaces. A linear map F : V1 → V2 is called a linear symplectomorphism if it is a linear isomorphism and (1) F ∗Ω2 = Ω1. Example. In Lecture 7 we have mentioned three simple linear symplectomorphisms f : (R 2n , Ω0) → (R 2n , Ω0): • f(x, ξ) = (−ξ, x). • f(x, ξ) = (x, ξ + Cx), where C is a symmetric n × n matrix. • f(x, ξ) = (Ax,(AT ) −1x), where A is an invertible n × n matrix. In fact, one can prove that any linear symplectomorphism is a composition of these simple ones. 1Recall that a linear 2-form is a anti-symmetric bilinear map, namely Ω(u, v) = −Ω(v, u). It it non-degenerate if Ω(u, v) = 0, ∀v ∈ Ω =⇒ u = 0. Equivalently, the induced map Ω : e V → V ∗ , Ω( e u)(v) = Ω(u, v) is bijective. 1
2 LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND Of course linear symplectomorphism defines an equivalent relation between symplectic vector spaces. It turns out that up to linear symplectomorphism, (R 2n , Ω0) is the only 2n-dimensional symplectic vector space: Theorem 1.3 (Linear Darboux theorem). For any linear symplectic vector space (V, Ω), there exists a linear symplectomorphism F : (V, Ω) → (R 2n , Ω0). Equivalently : given any symplectic vector space (V, Ω), there exists a dual basis {e ∗ 1 , · · · , e∗ n , f ∗ 1 , · · · , f ∗ n} of V ∗ so that as a linear 2-form, (2) Ω = Xn i=1 e ∗ i ∧ f ∗ i . The basis is called a Darboux basis of (V, Ω). Proof. Apply the Gram-Schmidt process with respect to the linear 2-form Ω. (For details, c.f. A. Canas de Silver, Lectures on Symplectic Geometry, page 1.) Remark. As a consequence, any symplectic vector space is even-dimensional. Since a linear symplectic form is a linear 2-form, a natural question is: which 2-form in Λ2 (V ∗ ) is a linear symplectic form on V ? Proposition 1.4. Let V be a 2n dimensional vector space. A linear 2-form Ω ∈ Λ 2 (V ∗ ) is a linear symplectic form on V if and only if as a 2n-form, (3) Ωn = Ω ∧ · · · ∧ Ω 6= 0 ∈ Λ 2n (V ∗ ). Proof. If Ω is symplectic, then according to the linear Darboux theorem, one can choose a dual basis of V ∗ so that Ω is given by (2). It follows Ω n = n!e ∗ 1 ∧ f ∗ 1 ∧ · · · ∧ e ∗ n ∧ f ∗ n 6= 0. Conversely, if Ω is degenerate, then there exists u ∈ V so that Ω(u, v) = 0 for all v ∈ V . Extend u into a basis {u1, · · · , u2n} of V with u1 = u. Then since dim Λ2n (V ) = 1, u1 ∧ · · · ∧ u2n is a basis of Λ2n (V ). But Ωn (u1 ∧ · · · ∧ u2n) = 0. So Ω n = 0. ¶ Symplectic Manifolds: Definitions and examples. Definition 1.5. A symplectic manifold is a pair (M, ω), where M is a smooth manifold, and ω ∈ Ω 2 (M) is a smooth 2-form on M, such that (1) for each p ∈ M, ωp ∈ Λ 2 (T ∗ p M) is a linear symplectic form on TpM. (2) ω is a closed 2-form, i.e. dω = 0 We call ω a symplectic form on M.
LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND 3 Remark. Note that if (M, ω) is symplectic, then dim M = dim TpM must be even. Denote dim M = 2n. Then as we have seen, 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. We will call ω n n! the symplectic volume form or the Liouville form on M. As a simple consequence of the existence of a volume form, we see M must be orientable. (There are many other topological restriction for the existence of a symplectic structure. For example, S 2n (n ≥ 1) does not admits any symplectic structure. In general it is very non-trivial to determine whether a manifold admits a symplectic structure. ) Still, we have plenty of interesting symplectic manifolds. 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 obviously (S, ω) is symplectic. Example. Let X be any smooth manifold and M = T ∗X its cotangent bundle. We will see below that there exists a canonical symplectic form ωcan on M. So, we have “as many” symplectic manifolds as smooth manifolds! ¶ The canonical symplectic structure on cotangent bundles. Let X be an n-dimensional smooth manifold and M = T ∗X its cotangent bundle. Let π : T ∗X → X, (x, ξ) 7→ x be the bundle projection map. From any coordinate patch (U, x1, · · · , xn) of X one can construct a system of coordinates (x1, · · · , xn, ξ1, · · · , ξn) on MU = π −1 (U). Namely, if ξ ∈ T ∗ xX, then ξ = Xξi(dxi)x. Using the computations at the beginning of Lecture 24, one can easily see that (4) ω := Xn i=1 dxi ∧ dξi is well-defined and is a symplectic form on M = T ∗X. Here is a coordinate free way to define ω: For any p = (x, ξ) ∈ M, we let (5) αp = (dπp) T ξ. Note that by definition ξ ∈ T ∗ xX, so for any p ∈ T ∗X, αp = (dπp) T ξ ∈ T ∗ p (T ∗X). In other words, we get a (globally defined) smooth 1-form α ∈ Ω 1 (M) = Γ∞(T ∗ (T ∗X)). Definition 1.6. We call α the canonical 1-form (or tautological 1-form) on T ∗X
4 LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND Proposition 1.7. In local coordinates described above, (6) α = Xn i=1 ξidxi . Proof. Let vp = Pn i=1(ai ∂ ∂xi + bi ∂ ∂ξi ) ∈ TpM. Then hαp, vpi = hξ,(dπp)vpi = h Xξi(dxi)x, Xai ∂ ∂xi i = Xaiξi = h Xξidxi , vpi. The equation follows. As a consequence, if we let (7) ω = −dα, then ω is closed, and is a symplectic form on M locally given by (4). ω = Xdxi ∧ dξi Definition 1.8. We call ω = −dα the canonical symplectic form on M = T ∗X. A crucial property for the canonical 1-form α ∈ Ω 1 (M) is the following Theorem 1.9 (Reproducing property). For any 1-form µ ∈ Ω 1 (X), if we let sµ : X → T ∗X be the map that sends x ∈ X to µx ∈ TxX, then we have (8) s ∗ µα = µ. Conversely, if α ∈ Ω 1 (M) is a 1-form such that (8) 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) T ξ. So at p = sµ(x) = (x, µx) we have αp = (dπp) T µx. It follows (s ∗ µα)x = (dsµ) T xαp = (dsµ) T x (dπp) T µx = (d(π ◦ sµ))T x µx = 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) T 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.
LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND 5 ¶ Symplectomorphisms. As in the linear case, we can define Definition 1.10. 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 (9) f ∗ω2 = ω1. We have the following amazing theorem for symplectic manifolds, whose proof can be found in A. Canas de Silver’s book mentioned above: Theorem 1.11 (Darboux theorem). Let (M, ω) be a symplectic manifold of dimension 2n. Then for any p ∈ M, there exists a coordinate patch (U, x1, · · · , xn, ξ1, · · · , ξn) centered at p such that on U, ω = Xdxi ∧ dξi . The coordinate patch above is called a Darboux coordinate patch. Remark. Equivalently, this says that one can find a neighborhood U near any point p ∈ M so that (U, ω) is symplectomorphic to (U, Ω0), where U is some open neighborhood of 0 in R 2n . So unlike Riemannian geometry, for symplectic manifolds there is no local geometry: locally all symplectic manifolds of the same dimension look the same. (However, there are much to say about the global geometry/topology of symplectic manifolds!) Remark. For cotangent bundle M = T ∗X with the canonical symplectic form, we have seen that any coordinate patch on X gives a Darboux coordinate patch on M. ¶ 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. According to our computations at the beginning of Lecture 18, we can “lift” f to a map ˜f : T ∗X → T ∗Y by (10) ˜f(x, ξ) = (f(x),(df T x ) −1 (ξ)). Theorem 1.12 (Naturality). The map ˜f : T ∗X → T ∗Y constructed above is a symplectomorphism with respect to the canonical symplectic forms. 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 T ◦ (dπT 2 )η = (dπT 1 ◦ df T )η = (dπT 1 )ξ = αT ∗X. This of course implies ˜f ∗ωT ∗Y = ωT ∗X.
6 LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND ¶ Hamiltonian theory. Now suppose (M, ω) is a symplectic manifold and H ∈ C ∞(M) a smooth function. Then dH ∈ Ω 1 (M) is a smooth 1-form on M. According to the non-degeneracy of ω, one can find a smooth vector field ΞH, called the Hamiltonian vector field on M so that (11) ιΞH ω = dH. This gives an intrinsic definition of ΞH that we defined for T ∗X via locally coordinates in Lecture 24. The following properties are immediate. Proposition 1.13. For any smooth function H ∈ C ∞(M), (1) LΞH H = 0, (2) LΞH ω = 0. Proof. We use the Cartan’s magic formula LX = dιX + ιXd: (1) LΞH H = ιΞH dH = ιΞH ιΞH ω = ω(ΞH, ΞH) = 0. (2) LΞH ω = dιΞH ω = ddH = 0. Let ρt : M → M be the Hamiltonian flow generated by ΞH. Then we get d dtρ ∗ t H = ρ ∗ tLΞH H = 0, i.e. ρ ∗ t H = H. So the Hamiltonian H is invariant under the Hamilton flow of H. This is just the conservation law in the abstract symplectic setting. We can do similar computation use the second one, d dtρ ∗ tω = ρ ∗ tLΞH ω = 0, i.e. ρ ∗ tω = ω. In other words, we proved Theorem 1.14. For any t, ρt : M → M is a symplectomorphism. As a consequence, as we mentioned in Lecture 24, the Liouville volume form is invariant under the Hamiltonian flow ρt . 2. Lagrangian submanifolds ¶ Linear subspaces in symplectic vector space. Now we study interesting linear subspaces of a symplectic vector space (V, Ω). Definition 2.1. The symplectic ortho-complement of a vector subspace W ⊂ V is (12) WΩ = {v ∈ V | Ω(v, w) = 0 for all w ∈ W}. Example. If (V, Ω) = (R 2n , Ω0) and W = span{e1, e2, f1, f3}, then WΩ = span{e2, f3, e4, · · · , en, f4, · · · , fn}.
LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND 7 One can easily observe the difference the symplectic ortho-complement and the standard ortho-complement W⊥ with respect to an inner product on V . For example, one always have W ∩ W⊥ = {0} while in most cases W ∩ WΩ 6= {0}. However, WΩ and W⊥ do have the same dimensions: Proposition 2.2. dim WΩ = 2n − dim W. Proof. Let Wf = Im(Ωe|W ) ⊂ V ∗ . Then dim Wf = dim W. But we also have WΩ = (Wf) 0 = {u ∈ V : l(u) = 0 for all l ∈ Wf}. So the conclusion follows. Definition 2.3. A vector subspace W of a symplectic vector space (V, Ω) is called • isotropic if W ⊂ WΩ. – Equivalently: Ω|W×W = 0. – Equivalently: ι ∗Ω = 0 ∈ Λ 2 (W∗ ), where ι : W ,→ V is the inclusion. – In particular dim W ≤ dim V/2. • coisotropic if W ⊃ WΩ. – Equivalently: WΩ is isotropic. – In particular dim W ≥ dim V/2. • Lagrangian if W = WΩ. – Equivalently: W is isotropic and dim W = dim V/2. – Equivalently: W is coisotropic and dim W = dim V/2. – Equivalently: W is both isotropic and coisotropic. – In particular dim W = dim V/2. • symplectic if W ∩ WΩ = ∅. – Equivalently: Ω|W×W is a linear symplectic form on W. – Equivalently: WΩ is a symplectic subspace. – In particular dim W is even. Example. If {e1, · · · , en, f1, · · · , fn} is a Darboux basis of (V, Ω), then for any 0 ≤ k ≤ n, Wk = span{e1, · · · , ek, fk+1, · · · , fn} is a Lagrangian subspace of (V, Ω). Example. Let F : (V1, Ω1) → (V2, Ω2) be any linear symplectomorphism. Note that Ω = Ω1 ⊕ (−Ω2) is a symplectic structure on V = V1 ⊕ V2. It is easy to check that the graph of F, Γ = {(v1, F(v1)) | v1 ∈ V1}, is a Lagrangian subspace of (V, Ω). ¶ Lagrangian submanifolds. Similarly for symplectic manifolds, we can define Definition 2.4. Let (M, ω) be a 2n-dimensional symplectic manifold and Z ⊂ M a submanifold. We call Z • isotropic if for all p ∈ Z, TpZ is an isotropic subspace of (TpM, ωp).
8 LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND • 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). One of the most important objects for us are the Lagrangian submanifolds. Even for the case of (R 2n , Ω0), Lagrangian submanifolds (not just linear Lagrangian subspaces) are fundamental objects and plays a crucial role in microlocal analysis. By definition, a Lagrangian submanfold Γ of (T ∗X, ωcan) has dimension dim Γ = dim X and satisfies ι ∗ωcan = 0, where ι : Γ ,→ T ∗X is the inclusion map. From the local expression of ω, it is easy to see the following are Lagrangian submanifolds: 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. Example. The zero section of T ∗X, X0 = {(x, ξ) ∈ T ∗X | x ∈ X, ξ = 0 ∈ T ∗ xX} is a (horizontal) Lagrangian submanifold of T ∗X. Example. For any submanifold S of X, the conormal bundle N ∗S = {(x, ξ) ∈ T ∗X | x ∈ S, ξ ∈ T ∗ xX so that ξ(v) = 0 for all v ∈ TxS} is a Lagrangian submanifold of T ∗X. Note: the previous two examples are special cases of this example. ¶ Horizontal Lagrangian submanifolds. Let M = T ∗X be the cotangent bundle of any smooth manifold X and ω the canonical symplectic form on M. We have seen that the zero section X0 = {(x, 0)} is a horizontal Lagrangian submanifold of M. A natural question is: what are other horizontal Lagrangian submanifolds? Of course, an n dimensional submanifold of M = T ∗X is horizontal means it is the graph of a section sµ : X → T ∗X. In other words, any horizontal submanifold has the form Λµ = {(x, µx) | x ∈ X}, where µ ∈ Ω 1 (X) is a smooth 1-form. When will this be Lagrangian? Let ι : Λµ ,→ T ∗X be the inclusion map. Note that π ◦ ι : Λµ → X is a diffeomorphism. Let γ : X → Λµ be its inverse. Then by definition sµ = ι ◦ γ. So from the reproducing property, Λµ is Lagrangian ⇔ ι ∗ dα = 0 ⇔ γ ∗ ι ∗ dα = 0 ⇔ d(s ∗ µα) = 0 ⇔ dµ = 0. In other words, we proved Proposition 2.5. A horizontal submanifold Λµ is Lagrangian if and only if dµ = 0
LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND 9 Definition 2.6. If µ is exact, i.e. µ = dϕ for some smooth function ϕ ∈ C ∞(X), then we call ϕ a generating function of the Lagrangian submanifold Λµ. ¶ Lagrangian submanifolds v.s. symplectomorphisms. Next let’s study the relation between Lagrangian submanifolds and symplectomorphisms. Let (M1, ω1) and (M2, ω2) be 2n dimensional symplectic manifolds. Then on the product M = M1 × M2 one has two important symplectic forms: • the product symplectic form ω = ω1 ⊕ ω2, • the twisted product form ωe = ω1 ⊕ (−ω2). Let f : M1 → M2 be a diffeomorphism, then its graph Γf = {(x, f(x) | x ∈ M1} is a 2n dimensional submanifold of the 4n dimensional manifold M. Theorem 2.7. f is a symplectomorphism if and only if Λf is Lagrangian with respect to ωe. Proof. Let ι : Γf ,→ M be the inclusion and γ : M1 → Γf ⊂ M be the diffeomorphism onto Γf , then Γf is Lagrangian ⇐⇒ ι ∗ωe = 0 ⇐⇒ γ ∗ ι ∗ωe = 0 ⇐⇒ ω1 − f ∗ω2 = 0. In particular, suppose M1 = T ∗X1 and M2 = T ∗X2 be cotangent bundles and ω1 = −dα1, ω2 = −dα2 the canonical symplectic forms. Then M = M1×M2 = T ∗X, where X = X1×X2. Moreover, the canonical 1-form on M = T ∗X is α = α1⊕α2, so the product symplectic form ω = ω1 ⊕ ω2 on M = T ∗X is the canonical symplectic form. Let σ2 : M2 → M2,(x, ξ) 7→ (x, −ξ). Then σ ∗ 2α2 = −α2, and thus σ ∗ 2ω2 = −ω2. It follows from the previous theorem that Proposition 2.8. If f : M1 → M2 is a diffeomorphism, then f is a symplectomorphism if and only if Γσ2◦f is a Lagrangian submanifold of (M, ω). ¶ Generating functions for symplectomorphisms. Note that proposition 2.8 is equivalent to The graph of f is a Lagrangian ⇐⇒ σ2 ◦ f is a symplectomorphism. From this correspondence it is natural to define Definition 2.9. If Γf = Λdϕ for some ϕ ∈ C ∞(X1 × X2), we say ϕ a generating function for the symplectomorphism σ2 ◦ f. Remark. Usually one only need to find generating functions locally.
10 LECTURE 26: FIO – SYMPLECTIC GEOMETRY BACKGROUND Now suppose we have a Lagrangian submanifold Λdϕ ⊂ T ∗ (X1 × X2) generated by function ϕ ∈ C ∞(X1 × X2). When will it generate a symplectomorphism σ2 ◦ f? In other words, we want Λdϕ to be the graph of some diffeomorphism f : M1 → M2. We denote pri : M = M1 × M2 → Mi be the projection maps. We choose local coordinate patches (U1, x1, · · · , xn) and (U2, y1, · · · , yn) on X1 and X2 respectively. Then Λdϕ is described locally by the equations ξi = ∂ϕ ∂xi , ηi = ∂ϕ ∂yi . Therefore, given any point (x, ξ) ∈ M1, a point (y, η) is the image of (x, ξ) under the map f if and only if (x, y, ξ, −η) is a point on the graph Λdϕ. So to find (y, η) = f(x, ξ), we need to solve the equations (13) ( ξi = ∂ϕ ∂xi (x, y), ηi = − ∂ϕ ∂yi (x, y). According to the implicit function theorem, to solve the first equation ξi = ∂ϕ ∂xi (x, y) for y locally, we need the condition (14) det ∂ 2ϕ ∂xi∂yj 6= 0. Of course after solving y we may feed it into the second equation to get η. Example. Let X1 = X2 = R n and B = (bij ) a non-singular n × n matrix. Then the function ϕ(x, y) = Pbijxiyj generates a linear symplectomorphism TB : T ∗R n → T ∗R n which maps (x, ξ) to (B−1 ξ, −BT x). In particular, if B = I, i.e. ϕ(x, y) = Pxiyi , then TB maps (x, ξ) to (ξ, −x). Example. Let X1 = X2 = R n and ϕ(x, y) = − |x−y| 2 2 . Then equation (13) becomes ( ξi = ∂ϕ ∂xi (x, y) = yi − xi ηi = − ∂ϕ ∂yi (x, y) = yi − xi ⇔ yi = xi + ξi , ηi = ξi . So the symplectomorphism generated by ϕ is f(x, ξ) = (x + ξ, ξ). More generally, if X is a Riemannian manifold and ϕ(x, y) = − d(x,y) 2 2 , where d(x, y) is the Riemannian distance from x to y, then the symplectomorphism generated by ϕ is the geodesic flow. Unfortunately not all Lagrangian submanifolds admits a generating function as described above. We will extend the conception of generating function by introducing “auxiliary variables” so that every Lagrangian submanifold of T ∗X is locally represented by a generating function