LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUND1.SYMPLECTIC STRUCTUREON COTANGENT BUNDLELinear symplectic structure.Definition 1.1. A symplectic vector space is a pair (V,2), where V is a real vectorspace, and 2 :V × V -→ R a non-degenerate linear 2-form.1 2 is called a linearsymplectic structure or a linear symplectic form on V.Erample. Let V =R2n - Rn × Rn and define2o((r,s), (y,n)) := (r, n)- (s,y)then (V, 2o) is a symplectic vector space. Let [ei,..,en, fi,..-, fn) be the stan-dard basis of R" × R", and (e, ... ,en, fi,... , fn) the dual basis of (R")* × (R")*,then as a linear 2-form one hasNo =Ee, ^ fi. Linear Darboux theorem.Definition1.2.Let (Vi,2)and (V2,22)be symplectic vector spaces.A linear mapF:Vi→V2iscalled a linear symplectomorphismif itisa linearisomorphismand(1)F*22=21.Erample.In Lecture7we have mentioned three simplelinear symplectomorphismsf : (R2n, 20) → (R2n,20):. f(r,) =(-E,r).. f(r,) = (r, + Cr), where C is a symmetric n × n matrix.. f(r,)= (Ar,(AT)-lc), where A is an invertible n × n matrixIn fact, one can prove that any linear symplectomorphism is a composition of thesesimple ones.1Recall that a linear 2-form is a anti-symmetric bilinear map, namely 2(u, u) =-2(u, u). It itnon-degenerate if2(u,v)=0,VvEQu=0Equivalently, the induced map2: V→V*, 2(u)(u) =2(u,v)is bijective.1
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
2LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUNDOf course linear symplectomorphism defines an equivalent relation between symplectic vector spaces.It turns out that up to linear symplectomorphism,(R2n,2o)is the only 2n-dimensional symplectic vector space:Theorem 1.3 (Linear Darboux theorem).For any linear symplectic vector space(V,2), there erists a linear symplectomorphismF : (V,2) → (R2n,20)Equivalently : given any symplectic vector space (V,2), there erists a dual basis[ei, ...,en, fi, .. , fn} of V* so that as a linear 2-form,Q=ei^fi.(2)i=1The basis is called a Darboux basis of (V,2)Proof. Apply the Gram-Schmidt process with respect to the linear 2-form 2. (For口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:which2-form in A?(V*)is a linear symplectic form on V?Proposition 1.4. Let V be a 2n dimensional vector space.A linear 2-form EA2(V*) is a linear symplectic form on V if and only if as a 2n-form,(3)2"=2A..A2+0EA2n(V*).Proof.If 2 is symplectic, then according to the linear Darboux theorem, one canchoose a dual basis of V* so that 2 is given by (2). It followsQn=nler^fiA..AeAf+0.Conversely, if is degenerate, then there exists u V so that 2(u,) = o forall V. Extend u into a basis ui,*.., u2n] of V with ui = u. Then sincedim A2n(V) = 1, u1 N... N u2n is a basis of A2n(V). But S"n(ui ..- N u2n) = 0. So口S2n=0.SymplecticManifolds:Definitionsand examples.Definition 1.5. A symplectic manifold is a pair (M,w), where M is a smoothmanifold, and w E 2?(M) is a smooth 2-form on M, such that(1) for each p E M, wp E A2(T,M) is a linear symplectic form on T,M(2) w is a closed 2-form, i.e. dw = 0We call w a symplectic form on M
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.
3LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUNDRemark. Note that if (M,w) is symplectic, then dim M = dim T,M must be evenDenote dimM =2n.Then aswe have seen,wp0A2n(TM),i.e. wn is a non-vanishing 2n form, thus a volume form on M. We will call w" thesymplectic volume form or the Liouville form on M. As a simple consequence of theexistence of a volume form, we see M must be orientable. (There are many othertopological restriction for the existence of a symplectic structure. For example, 2n(n≥1)does not admits any symplectic structure. In general it is very non-trivialto determine whether a manifold admits a symplectic structure.)Still, we haveplenty of interesting symplectic manifolds.Erample.(R2n,2o) is of course the simplest symplectic manifold.Erample. Let S be any oriented surface and w any volumeform on S, then obviously(S,w) is symplectic.Erample. Let X be any smooth manifold and M - T*X its cotangent bundle. Wewill see below that there exists a canonical symplectic form wcan 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 bundleLetT:T*X →X, (C,)H→be the bundle projection map.From any coordinate patch (u, ri, ..., n) of Xone can construct a system of coordinates (ai,..,n,Si,.,Sn) on Mu=π-1(u).Namely, if e T+X, thenE=si(dai)r:Using the computations at the beginning of Lecture 24, one can easily see thatw :-driA dei(4)i=1is well-defined and is a symplectic form on M =T*X.Here is a coordinate free way to define w: For any p = (r,) e M, we let(5)Qp=(dp)TENote that by definition E e T*X, so for any p E T*X.ap = (dp)TE E T,(T*X).Inotherwords,wegeta(globallydefined)smooth1-formα E2'(M) = T(T*(T*X)),Definition 1.6. We call α the canonical 1-form (or tautological 1-form) on T*X
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
4LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUNDProposition 1.7. In local coordinates described above,TE(6)sidaiα=i=1Proof. Let Up = Dr=i(ai, + b最)e T,M. Then(ap,p)=(5,(dp)p)=(Es(dai)a,a)=Eaiti=(Esidti, p).口Theequation follows.As a consequence, if we let(7)w=-da,then w is closed, and is a symplectic form on M locally given by (4)w=driAdsiDefinition 1.8. We call w = -da the canonical symplectic form on M = T*XA crucial property for the canonical 1-form Q E '(M) is the followingTheorem 1.9 (Reproducing property).For any 1-form μ E '(X), if we let sμ:X-T*X be the map that sendsr EX to μrETX,then we have(8)sta=μ.Conversely, if α E 2'(M) is a 1-form such that (8) hold for all 1-form μ E '(X),then α is the canonical 1-form.Proof. At any point p = (r, 3) we have ap = (dip)Te. So at p = sμ(r) = (r, μa) wehave Qp= (dp)Tμa.It follows()=(ds)p=(ds)(d)= (d(Su)=μr.Conversely, suppose o E 2'(M) is another 1-form on M satisfying the reproducingproperty above, then for any 1-form μ E '(X), we have s*(α- ao) = 0. So for anyUETX.0 = ((dsμ)T(α- Q0)p, U) = (α- Qo)p, (dsμ)r(u)For each p = (r,), the set of all vectors of the this form,((dsu)ru μE2'(X),μr=E,UETrX,)口span T,M, so we conclude that α = αo
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.
5LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUNDTSymplectomorphisms.As in the linear case, we can defineDefinition 1.10.Let (Mi,wi) and (M2,w2)be symplectic manifolds. A smoothmap f : Mi -→ M2 is called a symplectomorphism (or a canonical transformation) ifit is a diffeomorphism and(9)f*w2 = Wi.We have the following amazing theorem for symplectic manifolds, whose proofcan be found in A. Canas de Silver's book mentioned above:Theorem 1.11 (Darboux theorem). Let (M,w) be a symplectic manifold of dimen-sion 2n. Then for any p M, there erists a coordinate patch (u, i,..,En,Si,...,sn)centered at p such that on u,w=driΛdeiThe coordinate patch above is called a Darboux coordinate patch.Remark. Equivalently, this says that one can find a neighborhood u near any pointp E M so that (u,w) is symplectomorphic to (U,2o), where U is some open neigh-borhood of o in R2n.So unlike Riemannian geometry,for symplecticmanifolds thereis no local geometry: locally all symplectic manifolds of the same dimension lookthe same. (However, there are much to say about the global geometry/topology ofsymplectic manifolds!)Remark. For cotangent bundle M = T*X with the canonical symplectic form, wehave 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 nat-ural in the following sense: Suppose X and Y are smooth manifolds of dimension nand f :X →Y a diffeomorphism. According to our computations at the beginningof Lecture 18, we can“lift"f to a map f :T*X-→T*Y by(10)f(r,) = (f(r), (dfT)-1())Theorem 1.12 (Naturality). The map j : T*X -→ T*Y constructed above is asymplectomorphism with respect to the canonical symplectic forms.Proof. It is not hard to check that f is a diffeomorphism. Denote the projectionsby i : T*X → X and π2 : T*Y → Y. By definitionT2of=foT1.So if we denote f(r,s) = (y,n), then*T*=dfT(d2)=(ddfT)=(d)=QTx.口This of course implies f*wT+y =wT+x
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.
6LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUNDHamiltonian theory.Now suppose (M,w) isa symplectic manifold and H e Co(M)a smooth function. Then dH e '(M) is a smooth 1-form on M. According to the non-degeneracyof w, one can find a smooth vector field EH, called the Hamiltonian vector field onM so that(11)t=nW= dH.This gives an intrinsic definition of EH that we defined for T*X via locally coordi-nates in Lecture 24.Thefollowingpropertiesare immediate.Proposition1.13.Forany smooth functionHECo(M)(1) C=H = 0,(2) C=nW = 0. Proof. We use the Cartan's magic formula Cx = dix + txd:(1)C=H=H=E=(H,H)=0.口(2) C=nW= di=W = ddH = 0.Letpt:M-→MbetheHamiltonianflowgeneratedbyEh.ThenwegetdPH=PiCnH=0,i.e.ptH = 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 dosimilar computation use the second one,dPw = PiC=nw=0,i.e. ptw = w. In other words, we provedTheorem 1.14. For any t, pt : M → M is a symplectomorphism.As a consequence, as we mentioned in Lecture 24, the Liouville volume form isinvariant under the Hamiltonian flow Pt.2.LAGRANGIAN SUBMANIFOLDS Linear subspaces in symplectic vector space.Now we study interesting linear subspaces of a symplectic vector space (V,2)Definition 2.1. The symplectic ortho-complement of a vector subspace W C V is(12)W?- (eV/2(,w)=o forall wEW).Erample.If (V,2)= (R2n,2o)and W= span[ei,e2, fi, f3),thenW? spanfe2, fs, e4,*..,en, f4,..., fn]
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}.
LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUND7One can easily observe the difference the symplectic ortho-complement and thestandard ortho-complement Wl with respect to an innerproduct on V. For exam-ple, one always haveWnw+=[0] while inmost cases Wnw"+[0].However,w and w+ do have the same dimensions:Proposition 2.2. dimW= 2n - dim W.Proof. Let W - Im(2lw) c V*. Then dim W = dim W. But we also havewo=(w)o={uEV:l(u)=oforalllew)口So the conclusion follows.Definition 2.3. A vector subspace W of a symplectic vector space (V,2) is called. isotropic if W cw?.-Equivalently: 2/wxw- 0.-Equivalently: * = 0 e A?(W*), where t : W V is the inclusion.- In particular dim W ≤dimV/2.. coisotropic if wwn-Equivalently:w is isotropic.- In particular dim W ≥ dim V/2.. Lagrangian if W = W?.-Equivalently:W is isotropic and dimW=dimV/2.-Equivalently: W is coisotropic and dimW= dimV/2-Equivalently:Wis both isotropic and coisotropic- In particular dim W - dim V/2..symplectic if Wnw?-0.-Equivalently:2wxw is a linear symplecticform on W.-Equivalently:W is a symplectic subspace.- In particular dim W is even.Erample. If (ei,... , en, fi,..., fn] is a Darboux basis of (V, 2), then for any O ≤k ≤ n, W = span(e1,... ,ek, fu+1, .. , fn) is a Lagrangian subspace of (V,2)Erample.LetF:(Vi.2)→(V2.22)beanylinearsymplectomorphism.Notethat=2i@(-22) is a symplectic structure on V=Vi@V2. It is easy to check thatthe graph of F,F = [(U1, F(u) / Ui E Vi),is a Lagrangian subspace of (V, 2). Lagrangian submanifolds.Similarly for symplectic manifolds, we can defineDefinition 2.4.Let (M,w)be a 2n-dimensional symplectic manifold and Z C Ma submanifold. We call Z. isotropic if for all p E Z, TpZ is an isotropic subspace of (TpM,wp)
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).
8LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUND.coisotropic if for all p E Z, T,Z is a coisotropic subspace of (T,M,wp)..Lagrangian if for all p EZ,T,Z is a Lagrangian subspace of (T,M,wp). symplectic if for all p E Z, T,Z is a symplectic subspace of (T,M,wp).One of the most important objects for us are the Lagrangian submanifolds.Even for the case of (R2n, 2o), Lagrangian submanifolds (not just linear Lagrangiansubspaces)arefundamental objectsandplaysacrucialroleinmicrolocalanalysisBy definition, a Lagrangian submanfold I of (T* X, wcan) has dimension dim F :dim X and satisfies i*wean = 0, where : I T*X is the inclusion map. From thelocal expression of w, it is easy to see the following are Lagrangian submanifolds:Erample. For each r E X, the cotangent fiberT*X={(C,E)ET*XIEETX)is a (vertical) Lagrangian submanifold of T*X.Erample. The zero section of T*X,Xo={(C,)ET*X IREX,S=OETX)is a (horizontal) Lagrangian submanifold of T*X.Ecample.For any submanifold S of X, the conormal bundleN*S=((,E)ET*XESSETXsothatE(u)=OforallETS)is a Lagrangian submanifold of T*X.Note: the previous two examples are specialcases of this example. Horizontal Lagrangian submanifolds.Let M = T*X be the cotangent bundle of any smooth manifold X and w thecanonical symplectic form on M. We have seen that the zero section Xo=((r, O))is a horizontal Lagrangian submanifold of M. A natural question is: what are otherhorizontal Lagrangian submanifolds? Of course, an n dimensional submanifold ofM = T*X is horizontal means it is the graph of a section sμ : X - T*X. In otherwords, any horizontal submanifold has theformA=((,)EX)where μ e '(X) is a smooth 1-form. When will this be Lagrangian? Let t : A, →T*X be the inclusion map. Note that π o : Aμ → X is a diffeomorphism. Let:X-→Abe its inverse.Then by definitionSu=10Sofrom thereproducingproperty,A,isLagrangian台da=0台da=0台d(sa)=0台dμ=0Inotherwords,weprovedProposition 2.5. A horizontal submanifold A, is Lagrangian if and only if dμ=0
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
9LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUNDDefinition 2.6. If μ is exact, i.e. μ=dp for some smooth function E C(X),then we call a generating function of the Lagrangian submanifold Ay Lagrangian submanifolds v.s. symplectomorphisms.Next let's study the relation between Lagrangian submanifolds and symplec-tomorphisms. Let (Mi,wi) and (M2,w2) be 2n dimensional symplectic manifolds.Then on theproduct M =Mi× M2 onehas two importantsymplectic forms:.the product symplectic formw=wi w2,.thetwisted product form w=wi@(-w2)Letf :Mi→M2 be adiffeomorphism, then its graphIf= [(C, f(r) I rE M)isa2ndimensionalsubmanifoldofthe4ndimensionalmanifoldM.Theorem 2.7. f is a symplectomorphism if and only if Ay is Lagrangian withrespect to w.Proof. Let : Fy → M be the inclusion and : Mi -→ Iy C M be the diffeomor-phism onto If, thenI is Lagrangian *w=0*w=0wi-f*w2=0.口In particular, suppose Mi = T*Xi and M2 = T*X, be cotangent bundles andwi =-dai,w2= -da2 the canonical symplectic forms. Then M = Mi ×M2 =T*X,whereX=XixX2.Moreover,the canonical 1-form onM =T*X isα=αi@a2,sothe product symplectic form w = wi @ w2 on M = T*X is the canonical symplecticform. Let02 : M2 → M2, (r, $) -→ (r, -5)Then o2a2 = -Q2, and thus o2w2 = -w2. It follows from the previous theorem thatProposition 2.8. If f : Mi-→ M2 is a diffeomorphism, then f is a symplectomor-phism if and only if Fa2of is a Lagrangian submanifold of (M,w). Generating functions for symplectomorphisms.Note that proposition 2.8 is equivalent toThe graph of f is a Lagrangian 2 o f is a symplectomorphismFrom this correspondence it is natural to defineDefinition 2.9. If Iy =Adfor some E C(Xi × X2), we say a generatingfunctionforthesymplectomorphismo2of.Remark. Usually one only need to find generating functions locally
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.
10LECTURE26:FIO-SYMPLECTICGEOMETRYBACKGROUNDNow suppose we have a Lagrangian submanifold Adp C T*(Xi × X2) generatedby function e Co(Xi x X2).When will it generate a symplectomorphism g2 o f?In other words, we want Ad to be the graph of some diffeomorphism f : Mi → M2.We denote pri : M = Mi × M, -→ M, be the projection maps. We choose localcoordinate patches (Ur, &1,..., rn) and (U2, yi,... , yn) on Xi and X2 respectively.Then Adp is described locally by the equations s; = %, n: = %. Therefore, givenany point (r,s) Mi, a point (y,n) is the image of (r,s) under the map f if andonly if (c, y, s, -n) is a point on the graph Adp. So to find (y, n) = f(r, E), we needto solve the equations(a,y),(13)%(a,9).=aAccording to the implicit function theorem, to solve the first equation &; = (r,y)forylocally,we need the condition80det(14)￥0OrioyiOf course after solving y we may feed it into the second equation to get n.Erample. Let Xi = X, = Rn and B - (bi) a non-singular n × n matrix. Then thefunction (r,y) = bijriyj generates a linear symplectomorphism T : T*Rn -→T*Rn which maps (r, ) to (B-1s, -BTr).In particular, if B = I, i.e. (r, y) -riyi, then T maps (r,s) to (s, -r)Erample. Let Xi = X2 = R" and p(r,g) = --. Then equation (13) becomess=(,y)=yi-yi=ri+s,ar(n=-%(a,)=-Ini=si.So the symplectomorphism generated by is f(r,$) = (r + s,s).More generally, if X is a Riemannian manifold and p(r,3) = _(g, whered(,y) is the Riemannian distance from r to y, then the symplectomorphism gen-erated by is the geodesic flow.Unfortunately not all Lagrangian submanifolds admits a generating function asdescribed above. We will extend the conception of generating function by introduc-ing “auxiliary variables" so that every Lagrangian submanifold of T*X is locallyrepresented by a generating function
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