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.