Darboux s theorem an symplectic geometry Liang, Feng May 9, 2014 Abstract Symplectic geometry is a very important branch of ifferential geometry, it is a special case of poisson geometry, an coul also give some geometric view of classical mechanics. In this project, we will give some basic concepts an properties of Symplectic manifol; moreover, we will also introuce the very funamental property of Symplectic geometry- Darboux s theorem, which shows that any Symplectic manifol is locally isomorphic to some R 2n with its stanar symplectic form. 1 Basic concepts A Symplectic manifol is a smooth manifol M of even imension 2n with a symplectic structure: a nonegenerate close 2-form ω Ω 2 (M) Notice such ω satisfies that: ω(u, v) = ω(v, u) if ω(u, v) = 0 for v, then u = 0 We call a pair (V, ω) of a finite imensional real space with a bilinear form ω: V V R a Symplectic vector space if the ω satisfies the two conitions above. By such efinition, (T p M, ω q ) is a Symplectic vector space. Moreover, as ω close, it efines a homotopy class a = [ω] H 2 (M; R) Give some examples: Example1.1 1
Sympletic manifol R 2n with stanar symplectic form ω 0 = n x i y i i=1 Notice: ω 0 ω 0 is a volume form. Every Riemann surface with its area form is a symplectic manifol The prouct of two symplectic manifol M 1 M 2 is a symplectic manifol with the symplectic form ω 1 ω2 Remark ω gives a isomorphism: T M T M : X ι(x)ω = ω(x, ) Uner this isomorphism, we can consier more structures on a Sympletic manifol. A symplectomorphism of a symplectic manifol (M, ω) is a iffeomorphism ψ Diff(M) which preserves the symplectic form ω = ψ ω enote the group of symplectomorphisms by Symp(M, ω), or just Symp(M). A vector fiel X χ(m) is calle symplectic if ι(x)ω is close. Example1.2 ψ : R 4 R 2 ; (x, y) (x + j, A j y + k), A j = ( ) 1 j 2 0 1, (j, k) Z 2. It preserves the stanar syplectic form. In the course of ifferential manifol, we have seen that a vector fiel on a smooth manifol M can give a flow on it which is a family of iffeomorphisms in Diff(M); conversely, we can get a vector fiel by ifferentiating a family of iffeomorphisms. Inee, we can also get a similar corresponence between symplectomorphisms an symplectic vector fiels. It follows the next proposition Proposition 1.1 Let M be a close manifol. If t ψ t Diff (M) is a smooth family of iffeomorphisms generate by a family of vector fiels X t χ(m) via t ψ t = X t ψ t, ψ 0 = i, 2
then ψ t Symp (M, ω) for every t if an only if X t χ(m, ω) for every t. Moreover, if X, Y χ(m, ω) then [X, Y ] χ(m, ω) an ι([x, Y ])ω = H, where H = ω(x, Y ). Proof: (1) t ψ t ω = ψ t (ι(x t )ω + (ι(x t )ω)) = ψ t (ι(x t )ω), L X ω = ι(x)ω + (ι(x)ω) (2) ι([x, Y ])ω = t ι(ψt X)ω = L Y (ι(x)ω) = ι(y )ι(x)ω = (ω(x, Y )). t=0 As we mentione in the very beginning, to give a well relation with the classical mechanics, we also want to stuy the object relate to Energy, or more generally, Hamiltonian. For a Symplectic manifol (M, ω), consier a smooth function H : M R, we can etermine a vector fiel by: ι(x)ω = H We call such vector fiel the Hamiltonian vector fiel associate to the Hamiltonian function H. Also, efine the associate Hamiltonian flow φ t H Diff(M), which satisfies: t φt H = X H φ t H, φ 0 H = i Remark By noticing the fact: H(X H ) = (ι(x H )ω)(x H ) = ω(x H, X H ) = 0 can see that Hamiltonian vector fiel is along to the level set of Hamiltonian function H. Example1.3 H : S 2 R; (x 1, x 2, x 3 ) x 3, φ t H is the rotation along x 3-axis with constant spee, X H is simply θ 3
As expecte, there are some properties for Hamiltonian things. Proposition 1.2 (M, ω) symplectic manifol. i Hamiltonian flow φ t H Symp(M, ω), tangent to the level surfaces of H. ii H : M R an ψ Symp (M, ω), then X H ψ = ψ X H. iii [X F, X G ] = X {F,G}, Poisson bracket efine as F, G = ω(x F, X H ) = F (X H ) Proof : (i) irectly from last Proposition 3.2 an closeness from efinitions. (ii) comes from: ι(x H ψ )ω = (H ψ) = ψ H = ψι(x H )ω = ι(ψ X H )ω. (iii) By (i), φ t G Symp(M, ω), then [X F, X G ] = t (φ t F ) X G = t=0 t X G φ t F. t=0 Hence ι([x F, Y G ])ω = t (G φ t F ) t=0 = t G φ t F = (G(X F )) = {G, F } t=0 = {F, G} = X {F,G} After knowing some basic properties of Hamiltonian things. We want to exten some more concepts in ifferential geometry, give the efinition of Hmiltonian isotopies. In topology or ifferential geometry, Isotopy is a topology concepts, which is a family of embeings connecting 2 continuous embeings in a smooth manifol or a topological space. Then, relate it with Symlectic manifol. M Symplectic, consier smooth map [0, 1] M M : (t, q) ψ t (q) with ψ t Symp(M, ω) an ψ 0 =i. Call such a family of symplectomorphism symplectic isotopies. Given ψ t, we can fin its relate vector fiels X t by t ψ t = X t ψ t, X t) symplectic by previous proposition, shows ι(x t )ω close. Furthermore, if it is a bounary with ι(x t )ω=h, we call 4
such H t time-epenent Hamiltonian, call such ψ t Hamiltonian isotopy. For ψ Symp(M, ω) with a Hamiltonian isotopy ψ t connecting it an ientity, i.e. ψ o =i, ψ 1 = ψ, we call such ψ is Hamiltonian. Denote the group of Hamiltoniansymplectomorphisms by Ham(M, ω), it is normal in Symp(M). Example1.4 unknot is not isotopic to Trefoil knot. unknot is not isotopic to Trefoil knot. Before proving the Darboux s theorem, we want to give a cotangent boule T L a symplectic struction to finish the basic concept part. For a smooth M, let x = (x 1,..., x n ): U R n the local coorinate chart on L, as T q L has the basis x i, v T q L, it is in a unique form λ can := n i=1 y i x i, this gives the canonical 1-form, by taking the ifferential, we get the canonical 2-form ω can = λ can = x y. By this construction, we see that inee, canonical 1-form at (x, v ) is just v itself in some sense. Moreover, not har to check that (T L, ω can ) is a symplectic manifol, which means that we fin a symplectic structure for that cotangent bunle. There is also a proposition without proof here follows: Proposition 1.3 λ can is uniquely characterize by the property: σ λ can = σ, 1-form σ. 2 Darboux s theorem Finally, we come to a very important an funamental results in many fiels-darboux s theorem. It was foun by Jean Gaston Darboux when he solve the Pfaff problem. Here, 5
we will talk about it in the most known form for symplectic geometry. Darboux s theorem Every symplectic form ω on M is locally iffeomorphic to the stanar form ω 0 on R 2n This theorem tells us any symplectic manifol in same imension looks same locally. Next, we give a proof by applying the Moser s argument. Lemma 1/ Moser s argument For every family of symplectic forms ω t Ω 2 (M) which are exact:, t ω t = σ t then, a family of iffeomorphisms ψ t Diff(M) such that:. newline ψ t ω t = ω 0 Proof: We want to think ψ t as a flow of some X t, via ψ t t = X t ψ t, ψ 0 = i,, It satisfies the conition for the manifol, still nee to solve the equation to satisfies the conition for the symplectic form on it, ifferentiate both sie of ψ t ω t = ω 0, get: 0 = t ψ t ω t = ψ t ( t ω t + ι(x t )ω t + ι(x t )ω t ) = ψ t (σ t + ι(x t )ω t ) Solve: ψ t (σ t + ι(x t )ω t ), as ω t nonegenerate, get such X t. Move on to last Lemma for the Proof of Darboux theorem. Lemma 2 Let M be a 2n-imensional smooth manifol an Q M be a compact submanifol. Suppose that ω 0, ω 1 Ω 2 (M) are equal an non-egenerate on T q M. Then there exist open neighborhoos N 0 an N 1 of Q an a iffeomorphism ψ : N 0 N 1 such that ψ Q = i, ψ ω 1 = ω 0. This lemma is very strong. If we take Q as a point, this lemma tells us if two symplectic form are ientical or isomorphic at some point, they are ientical or isomorphic on their 6
germ. Also, we can see any symplectic form are isomorphic on M at a point, we can think locally M as R 2n, After this isomorphism, we are in R 2n. As we just point out, it then gives a isomorphism to the stanar form at a point, then, by lemma, we get an isomorphism to ω 0 on the germ of this point. Composite all germs, we get the Darboux s theorem. Proof of lemma 2: Proof: By Moser: enough to show some 1-form σ Ω 1 (N 0 ) such that σ TQ M= 0, σ = ω 1 ω 0. With ω t = ω 0 + t(ω 1 ω 0 ) = ω 0 + tσ satisfies the conition of Moser, on N 0. Shrinking N 0 to ensure that ω t is non-egenerate in N 0 for every t as Moser requires. Shrinking N 0 again to ensure the solution for Moser exist on the require time interval 0 t 1. Prove what we claim at the beginning of the proof: Consier the restriction of exponential map to the normal bunle T Q of the submanifol Q with respect to any Riemannian metric on M: exp : T Q M. Consier the neighbourhoo of the zero section: U ε = {(q, v) T M q Q, v T q Q, v < ε}. Then get a iffeomorphsim: N 0 = exp(u ε ) for ε > 0 sufficiently small. Define φ t : N 0 N 0 for 0 t 1 by φ t (exp(q, v)) = exp(q, tv). φ t is a iffeomorphism for t > 0 with φ 0 (N 0 ) Q, φ 1 = i, an φ t Q = i. This implies φ 0 τ = 0, φ 1 τ = τ, where τ = ω 1 ω 0. Then, can efine the vector fiel X t = ( t φ t) φ 1 t for t > 0. X t singular at t = 0. However, we obtain t φ t τ = φ t L Xt τ = (φ t ι(x t )τ) = σ t, but such family of σ t (q; v) = τ(φ t (q); t φ t(q), φ t (q)v) smooth at t = 0 an vanishes on Q. 7
Finally, we get: This finish the proof. τ = φ 1 τ φ 0 τ = 1 0 t φ t τt = σ, σ = 1 0 σ t t. This paper follows part of the reference text Chapter 3, gives a introuction to the symplectic geometry. Darboux s theorem is very funamental an also very important. We can not expect the math worl without it nowaays. Hope we can fin more such theorems in the future. The way to the future is not flat but we will go further. Thank you. 8
References [1] Dusa McDuff, Dietmar Salamon (1998), Introuction to Symplectic Topology, Clarenon University Press 9