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 1 space, and Ω : V × V ! R a non-degenerate linear 2-form. Ω is called a linear symplectic structure or a linear symplectic form on V . Example. Let V = R2n = Rn × Rn and define Ω0((x; ξ); (y; η)) := hx; ηi − hξ; yi; then (V; Ω0) is a symplectic vector space. Let fe1; ··· ; en; f1; ··· ; fng be the stan- n n ∗ ∗ ∗ ∗ n ∗ n ∗ dard basis of R × R , and fe1; ··· ; en; f1 ; ··· ; fng the dual basis of (R ) × (R ) , then as a linear 2-form one has n X ∗ ∗ Ω0 = ei ^ fi : i=1 { 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 2n 2n f :(R ; Ω0) ! (R ; 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; 8v 2 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 sym- 2n plectic vector spaces. It turns out that up to linear symplectomorphism, (R ; 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 2n F :(V; Ω) ! (R ; Ω0): Equivalently : given any symplectic vector space (V; Ω), there exists a dual basis ∗ ∗ ∗ ∗ ∗ fe1; ··· ; en; f1 ; ··· ; fng of V so that as a linear 2-form, n X ∗ ∗ (2) Ω = ei ^ fi : i=1 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 2 Λ2(V ∗) is a linear symplectic form on V if and only if as a 2n-form, (3) Ωn = Ω ^ · · · ^ Ω 6= 0 2 Λ2n(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 n ∗ ∗ ∗ ∗ Ω = n!e1 ^ f1 ^ · · · ^ en ^ fn 6= 0: Conversely, if Ω is degenerate, then there exists u 2 V so that Ω(u; v) = 0 for all v 2 V . Extend u into a basis fu1; ··· ; u2ng of V with u1 = u. Then since 2n 2n n dim Λ (V ) = 1, u1 ^ · · · ^ u2n is a basis of Λ (V ). But Ω (u1 ^ · · · ^ u2n) = 0. So n Ω = 0. { Symplectic Manifolds: Definitions and examples. Definition 1.5. A symplectic manifold is a pair (M; !), where M is a smooth manifold, and ! 2 Ω2(M) is a smooth 2-form on M, such that 2 ∗ (1) for each p 2 M, !p 2 Λ (Tp M) is a linear symplectic form on TpM. (2) ! is a closed 2-form, i.e. d! = 0 We call ! a symplectic form on M. LECTURE 26: FIO { SYMPLECTIC GEOMETRY BACKGROUND 3 Remark. Note that if (M; !) is symplectic, then dim M = dim TpM must be even. Denote dim M = 2n. Then as we have seen, n 2n ∗ !p 6= 0 2 Λ (Tp M); n !n i.e. ! is a non-vanishing 2n form, thus a volume form on M. We will call 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, S2n (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. 2n Example. (R ; Ω0) is of course the simplest symplectic manifold. Example. Let S be any oriented surface and ! any volume form on S, then obviously (S; !) 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 !can 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 −1 one can construct a system of coordinates (x1; ··· ; xn; ξ1; ··· ; ξn) on MU = π (U). Namely, if ξ 2 T ∗X, then x X ξ = ξi(dxi)x: Using the computations at the beginning of Lecture 24, one can easily see that n X (4) ! := dxi ^ dξi i=1 is well-defined and is a symplectic form on M = T ∗X. Here is a coordinate free way to define !: For any p = (x; ξ) 2 M, we let T (5) αp = (dπp) ξ: ∗ ∗ Note that by definition ξ 2 Tx X, so for any p 2 T X, T ∗ ∗ αp = (dπp) ξ 2 Tp (T X): In other words, we get a (globally defined) smooth 1-form α 2 Ω1(M) = Γ1(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, n X (6) α = ξidxi: i=1 Pn @ @ Proof. Let vp = (ai + bi ) 2 TpM. Then i=1 @xi @ξi X X @ X X hαp; vpi = hξ; (dπp)vpi = h ξi(dxi)x; ai i = aiξi = h ξidxi; vpi: @xi The equation follows. As a consequence, if we let (7) ! = −dα; then ! is closed, and is a symplectic form on M locally given by (4). X ! = dxi ^ dξi Definition 1.8. We call ! = −dα the canonical symplectic form on M = T ∗X. A crucial property for the canonical 1-form α 2 Ω1(M) is the following 1 Theorem 1.9 (Reproducing property). For any 1-form µ 2 Ω (X), if we let sµ : ∗ X ! T X be the map that sends x 2 X to µx 2 TxX, then we have ∗ (8) sµα = µ. Conversely, if α 2 Ω1(M) is a 1-form such that (8) hold for all 1-form µ 2 Ω1(X), then α is the canonical 1-form. T Proof. At any point p = (x; ξ) we have αp = (dπp) ξ. So at p = sµ(x) = (x; µx) we T have αp = (dπp) µx. It follows ∗ T T T T (sµα)x = (dsµ)x αp = (dsµ)x (dπp) µx = (d(π ◦ sµ))x µx = µx: 1 Conversely, suppose α0 2 Ω (M) is another 1-form on M satisfying the reproducing 1 ∗ property above, then for any 1-form µ 2 Ω (X), we have sµ(α − α0) = 0. So for any v 2 TxX, T 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, 1 f(dsµ)xv j µ 2 Ω (X); µx = ξ; v 2 TxX; g 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;!1) and (M2;!2) 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 !2 = !1: 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; !) be a symplectic manifold of dimen- sion 2n. Then for any p 2 M, there exists a coordinate patch (U; x1; ··· ; xn; ξ1; ··· ; ξn) centered at p such that on U, X ! = dxi ^ 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 2 M so that (U;!) is symplectomorphic to (U; Ω0), where U is some open neigh- borhood of 0 in R2n. 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.