LECTURE 4: SYMPLECTOMORPHISMS Contents 1. The group of Symplectomorphisms 1 2. Symplectomorphisms as Lagrangian submanifolds 4 3. Generating functions 7 4. The billiards 9 1. The group of Symplectomorphisms ¶ Diffeomorphisms v.s. vector fields. Let M be a smooth manifold and Vect(M) the set of all smooth vectors on M. It is well known that for any X, Y ∈ Vect(M), the Lie bracket [X, Y ] = XY − Y X ∈ Vect(M) and is bilinear, anti-symmetric and satisfies the Jacobi identity [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0. As a consequence, (Vect(M), [·, ·]) is an (infinitely dimensional) Lie algebra. What is the corresponding “Lie group”? Well, any smooth vector field X ∈ Vect(M) generates (at least locally) a oneparameter subgroup of diffeomorphisms of M ρt = exp(tX) : M → M, ρt(x) = γ X x (t), where γ X x is the integral curve of X starting at x. Conversely, given any one parameter subgroup ρt of diffeomorphisms on M, one gets a smooth vector field via X(x) = d dt t=0 ρt(x). So, the group of diffeomorphisms, Diff(M) = {ϕ : M → M | ϕ is a diffeomorphism}, is the “Lie group” whose Lie algebra is Vect(M). 1
2 LECTURE 4: SYMPLECTOMORPHISMS ¶ The group of symplectomorphisms. Now let (M, ω) be a symplectic manifold and Symp(M, ω) = {ϕ : M → M | ϕ is a symplectomorphism} the group of symplectomorphisms of (M, ω). This is a “closed” subgroup of Diff(M). Both Symp(M, ω) and Diff(M) are large in the sense that they are infinitely dimensional. Example. We have seen in lecture 2 that any diffeomorphism ϕ : X → X lifts to a symplectomorphism ˜ϕ : T ∗X → T ∗X via ϕ˜(x, ξ) = (ϕ(x),(dϕ∗ x ) −1 (ξ)). It is easy to check ϕ^2 ◦ ϕ1 = ˜ϕ2 ◦ ϕ˜1. So we get a natural group monomorphism Diff(X) → Symp(T ∗X, ωcan). Example. Here is another subgroup of Symp(T ∗X, ωcan): For any β ∈ Ω 1 (X), let Gβ : T ∗X → T ∗X be the diffeomorphism Gβ(x, ξ) = (x, ξ + βx). Lemma 1.1. Gβ is a symplectomorphism if and only if β is closed. Proof. We shall prove G∗ βαcan − αcan = π ∗β, which implies the conclusion. Recall the reproducing property of αcan: αcan is the unique 1-form on T ∗X such that for any µ ∈ Ω 1 (X), s ∗ µα = µ. It follows s ∗ µG ∗ βαcan = (Gβ ◦ sµ) ∗αcan = s ∗ µ+βαcan = µ + β = s ∗ µπ ∗β + µ. So by reproducing property again we see G∗ βαcan − π ∗β = αcan. As a consequence, we get another group monomorphism Z 1 (X) → Symp(T ∗X, ωcan), where Z 1 (X) is the space of closed 1-forms on X. ¶ Symplectic vector fields. Question: What is the “Lie algebra” of Symp(M, ω)? Well, since Symp(M, ω) is a “closed” subgroup of Diff(M), its Lie algebra should be a Lie subalgebra of (Vect(M), [·, ·]). Let’s try to find the condition for a smooth vector to be “symplectic”. A symplectic vector field X should be a vector field whose flow {ρt} consists of symplectomorphisms. In other words ρ ∗ tω = ω for all t. It follows 0 = d dtρ ∗ tω = ρ ∗ tLXω. Definition 1.2. A smooth vector field X is called symplectic if LXω = 0. Since ω is closed, the Cartan’s magic formula implies
LECTURE 4: SYMPLECTOMORPHISMS 3 Lemma 1.3. A vector field X on (M, ω) is symplectic if and only if ιXω is closed. The set of all symplectic vector fields on (M, ω) is denoted by Vect(M, ω). We need to check that if X, Y ∈ Vect(M, ω), so is [X, Y ]. In other words, (Vect(M, ω), [·, ·]) is a Lie sub algebra of (Vect(M), [·, ·]). This follows from Lemma 1.4. If X, Y are symplectic, ι[X,Y ]ω = d(−ω(X, Y )). Proof. By the definition of exterior differential, for any Z ∈ Vect(M), 0 = (dω)(X, Y, Z) = X(ω(Y, Z))−Y (ω(X, Z))+Z(ω(X, Y ))−ω([X, Y ], Z)+ω([X, Z], Y )−ω([Y, Z], X) 0 = (dιXω)(Y, Z) = Y (ω(X, Z)) − Z(ω(X, Y )) − ω(X, [Y, Z]), 0 = (dιY ω)(X, Z) = X(ω(Y, Z)) − Z(ω(Y, X)) − ω(Y, [X, Z]). Comparing the three equations we conclude −Z(ω(X, Y )) − ω([X, Y ], Z) = 0, or in other words, ι[X,Y ]ω = d(−ω(X, Y )). Remark. So the bracket [X, Y ] of two symplectic vector fields is better than being a symplectic vector field: ι[X,Y ]ω is not only closed, but in fact exact!. ¶ Hamiltonian vector fields. Definition 1.5. A vector field X is Hamiltonian if ιXω is exact. Remark. According to the lemma above, the space of all Hamiltonian vector fields is an ideal of Vect(M, ω). So if X is hamiltonian, then there exists a smooth function f ∈ C ∞(M) so that ιXω = df. Conversely, since ω is non-degenerate, for any f ∈ C ∞(M), there is a unique vector field Xf on M so that ιXfω = df. We will call Xf the Hamiltonian vector field associated to the Hamiltonian function f. The flow generated by Xf is called the Hamiltonian flow associated to f. Remark. Assume M is connected. Then two smooth functions define the same Hamiltonian vector field if and only if they differ by a constant. So as a vector space the space of Hamiltonian vector fields is isomorphic to C ∞(M)/R, which can also be identified with C ∞(M)0 = {f ∈ C ∞(M) | Z M f(x)dx = 0.} if M is compact.
4 LECTURE 4: SYMPLECTOMORPHISMS Example. On the 2-sphere S 2 with symplectic form ω = dθ∧dz, if we take f(z, θ) = z the height function, then Xf = ∂ ∂θ , and the Hamiltonian flow is the rotation about the vertical axis: ρt(z, θ) = (z, θ + t). Example. On (T 2 , dθ1 ∧ dθ2) the vector fields ∂ ∂θ1 and ∂ ∂θ1 are both symplectic but not Hamiltonian. We will return to Hamiltonian vector fields later. ¶ Hamiltonian symplectomorphisms. Recall that an isotopy is a family of diffeomorphisms ρt so that ρ0 = Id. If each ρt is a symplectomorphism, we call the isotopy a symplectic isotopy. It is easy to see that a symplectic isotopy is generated by a family of symplectic vector fields Xt with d dtρt = Xt(ρt). If each Xt is not only symplectic, but in fact Hamiltonian, then we call the isotopy a Hamiltonian isotopy. Definition 1.6. A symplectomorphism ϕ is called Hamiltonian symplectomorphism if there exists a Hamiltonian isotopy ρt such that ρ1 = ϕ. The space of Hamiltonian symplectomorphisms is denoted by Ham(M, ω). It turns out that Ham(M, ω) is a normal subgroup of Symp(M, ω) whose Lie algebra is the algebra of all Hamiltonian vector fields. Since any function (modulo constants) gives a family of Hamiltonian symplectomorphisms, we see that the group of symplectomorphisms is huge. 2. Symplectomorphisms as Lagrangian submanifolds ¶ Lagrangian submanifolds v.s. symplectomorphisms. Weinstein’s symplectic creed: EVERYTHING IS A LAGRANGIAN SUBMANIFOLD! In what follows we shall study symplectomorphisms according to this creed. Let (M1, ω1) and (M2, ω2) be 2n dimensional symplectic manifolds and let pri : M1 × M2 → Mi be the projection. We have seen in lecture 2 that for any nonzero real numbers λ1 and λ2, λ1pr∗ 1ω1 + λ2pr∗ 2ω2 is a symplectic form on the product M = M1 × M2. In particular, one has two important symplectic forms: • the product symplectic form ω = pr∗ 1ω1 + pr∗ 2ω2, • the twisted product form ω˜ = pr∗ 1ω1 − pr∗ 2ω2
LECTURE 4: SYMPLECTOMORPHISMS 5 Now 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 M1 × M2. Theorem 2.1. f is a symplectomorphism if and only if Γf is Lagrangian with respect to ω˜. Proof. Let ι : Γf ,→ M be the inclusion and γ : M1 → Γf be the obvious diffeomorphism, then Γf is Lagrangian ⇐⇒ ι ∗ω˜ = 0 ⇐⇒ γ ∗ ι ∗ω˜ = 0 ⇐⇒ γ ∗ ι ∗ (pr∗ 1ω1 − pr∗ 2ω2) = 0 ⇐⇒ ω1 − f ∗ω2 = 0 ⇐⇒ f is a symplectomorphism. ¶ Lift of smooth maps as Lagrangian submanifolds. 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 theorem that Proposition 2.2. 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). Now suppose g : X1 → X2 is a diffeomorphism. As we have seen, g lifts to a symplectomorphism ˜g : M1 → M2. Recall that g˜(x, ξ) = (y, η) ⇐⇒ y = g(x), ξ = (dgx) ∗ η. As a consequence, Γ σ g˜ = {(x, ξ, y, η) | y = g(x), ξ = −(dgx) ∗ η} is a Lagrangian submanifold of M = T ∗X. Here is another way to see this: Let Xg = {(x, g(x)) | x ∈ X1} ⊂ X = X1 × X2 be the graph of g, then Γσ g˜ = N∗Xg. In fact, we have a more general theorem: Theorem 2.3. Let X1, X2 be arbitrary smooth manifolds and g : X1 → X2 a smooth map, then the set Γ σ g˜ defined as above is exactly N∗Xg, and thus a Lagrangian submanifold of M = T ∗X.
6 LECTURE 4: SYMPLECTOMORPHISMS Proof. For any (x, g(x)) ∈ Xg, T(x,g(x))Xg = {(v, dgx(v)) | v ∈ TxX1}. By definition, N∗ (x,g(x))Xg is the subspace of T ∗ (x,g(x))X that annihilates T(x,g(x))Xg. so (ξ, η) ∈ N ∗ (x,g(x))Xg ⇐⇒ hξ, vi + hη, dgx(v)i = 0, ∀v ∈ TxX1 ⇐⇒ hξ + (dgx) ∗ η, vi = 0, ∀v ∈ TxX1 ⇐⇒ ξ = −(dgx) ∗ η. ¶ Fixed points of symplectomorphisms. The identity map Id : M → M is a symplectomorphism, and the corresponding Lagrangian submanifold of (M × M, ω˜) is the diagonal ∆ = {(x, x) | x ∈ M}. We can canonically identify ∆ with M. According to the Weinstein’s Lagrangian neighborhood theorem, there exists a neighborhood U of ∆ ' M in (M × M, ω˜), a neighborhood U0 of M in T ∗M and a symplectomorphism ϕ : U0 → U. Definition 2.4. We say a diffeomorphism f : M → M is C 1 close to Id if its graph Γf lies in U, which, under the map ϕ −1 , is the graph of a 1-form µ on M. Now let (M, ω) be a compact symplectic manifold and f ∈ Symp(M, ω) a symplectomorphism that is C 1 closed to the identity symplectomorphism. Since Γf is a Lagrangian submanifold of (M × M, ω˜) and ϕ is a symplectomorphism, ϕ −1 (Γf ) is a Lagrangian submanifold of (T ∗M, ωcan). So if it is the graph of a one-form µ, µ must be closed. Remark. By this way we get an identification of a C 1 neighborhood of Id with a neighborhood of 0 in the space of closed 1-forms. So the tangent space of Id in Symp(M, ω) can be identified with the space of closed 1-forms on M. This coincides with our earlier “Lie group-Lie algebra” observation, since symplectic vector fields are in one-to-one correspondence with closed 1-forms under ω. Theorem 2.5. Let (M, ω) be a compact symplectic manifold with H1 (M) = 0. Then any symplectomorphism of M which is sufficiently C 1 close to Id has at least two fixed points. Proof. Let f ∈ Symp(M, ω) is C 1 close to Id. Then under the map ϕ, the graph of f is identified with a closed one-form µ on M. Since H1 (M) = 0, one can find a smooth function h ∈ C ∞(M) so that µ = dh. Since M is compact, h admits at least two critical points (the maximum and the minimum). Obviously any critical point of h gives an intersection point of Γf with ∆, which yields a fixed point of f.
LECTURE 4: SYMPLECTOMORPHISMS 7 ¶ The Arnold conjecture. Conjecture 2.6 (Arnold, symplectomorphism version). Let (M, ω) be a compact symplectic manifold and f : M → M a Hamiltonian symplectomorphism. Then #(fixed points of f) ≥ minimal number of critical points of a Morse function on M. Conjecture 2.7 (Arnold, Lagrangian version). Let (M, ω) be a compact symplectic manifold, L ⊂ M a Lagrangian submanifold, and f : M → M a Hamiltonian symplectomorphism. Then #(L ∩ f(L)) ≥ minimal number of critical points of a Morse function on L. Note that by Morse theory, the minimal number of critical points of a Morse function on M is at least P i dim Hi (M). The conjecture is only proven in special cases via the theory of Floer homology. 3. Generating functions ¶ Generating function for horizontal Lagrangian submanifolds. Let M = T ∗X be the cotangent bundle of any smooth manifold X and ω the canonical symplectic form. We have seen that a horizontal submanifold Xµ = {(x, µx) | x ∈ X}, is Lagrangian if and only if dµ = 0. Definition 3.1. 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. Note that proposition 2.2 is equivalent to The graph of f is a Lagrangian ⇔ σ2 ◦ f is a symplectomorphism. From this correspondence it is natural to define Definition 3.2. 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. ¶ Constructing symplectomorphisms. Now suppose we have a Lagrangian submanifold Λdϕ generated by function ϕ. When will it generate a symplectomorphism? 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
8 LECTURE 4: SYMPLECTOMORPHISMS equations ξi = ∂ϕ ∂xi , ηi = ∂ϕ ∂yi . Therefore, given any point (x, ξ) ∈ M1, to find its image (y, η) = f(x, ξ) we need to solve the equations (1) ( 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 (2) det ∂ 2ϕ ∂xi∂yj 6= 0. Of course after solving y we may feed it into the second equation to get η. ¶ Examples of generating functions. 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 (1) 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. Example. Let O be an open subset of R n × (Rn ) ∗ and ϕ = ϕ(x, η) ∈ C ∞(O) be a twisted generating function. Suppose det( ∂ 2ϕ ∂xi∂ηj ) 6= 0. Then by the same argument above or by composing the symplectomorphism we solved from (1) with the symplectomorphism (x, ξ) → (ξ, −x), we see that locally the set defined by ξi = ∂ϕ ∂xi (x, η), yi = ∂ϕ ∂ηi (x, η) is the graph of a symplectomorphism. Example. The identity symplectomorphism Id : T ∗R n → T ∗R n cannot be generated by functions of the usual form. However, if we take a twisted generating function ϕ(x, η) = Pxiηi , then it generates the identity symplectomorphism. Example. More generally, if U1 is an open subset of R n and f : U1 → U2 a diffeomorphism, then we have seen that its canonical lifting ˜f : T ∗U1 → T ∗U2 is a symplectomorphism. One can check that this is generated by ϕ(x, η) = Pfi(x)ηi
LECTURE 4: SYMPLECTOMORPHISMS 9 4. The billiards Student presentation