By Dixon E. T.
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Taking S = X, the conormal bundle L = X0 is the zero section of T ∗ X. 6 Lagrangian Complements Normal neighborhoods of lagrangian submanifolds are described by the theorems in the following two sections. , X is a submanifold lagrangian for ω0 and ω1 . We need some algebra for the Weinstein theorem. Suppose that U and W are n-dimensional vector spaces, and Ω : U ×W → R is a bilinear pairing; the map Ω gives rise to a linear map Ω : U → W ∗ , Ω(u) = Ω(u, ·). Then Ω is nondegenerate if and only if Ω is bijective.
HAMILTONIAN FIELDS Claim. , ρ∗t ω = ω, ∀t. Proof. We have d ∗ dt ρt ω = ρ∗t LXH ω = ρ∗t (d ıXH ω +ıXH dω ) = 0. dH 0 Therefore, every function on (M, ω) gives a family of symplectomorphisms. Notice how the proof involved both the nondegeneracy and the closedness of ω. 1 A vector field XH as above is called the hamiltonian vector field with hamiltonian function H. Example. The height function H(θ, h) = h on the sphere (M, ω) = (S 2 , dθ ∧ dh) has ∂ ıXH (dθ ∧ dh) = dh ⇐⇒ XH = . ∂θ Thus, ρt (θ, h) = (θ + t, h), which is rotation about the vertical axis; the height function H is preserved by this motion.
2. There exist neighborhoods U0 and U1 of X in N0 and a diffeomorphism θ : U0 → U1 such that θ U0 ✲ U1 ✒ i0 ❅ ■ ❅ ❅ i0 ❅ ❅ commutes and θ ∗ ω1 = ω 0 . X Take ϕ = ψ ◦ θ and U = ϕ(U0 ). Check that ϕ∗ ω = θ∗ ψ ∗ ω = ω0 . ω1 Remark. 14 classifies lagrangian embeddings: up to symplectomorphism, the set of lagrangian embeddings is the set of embeddings of manifolds into their cotangent bundles as zero sections. The classification of isotropic embeddings was also carried out by Weinstein in [45, 46]. An isotropic embedding of a manifold X into a symplectic manifold (M, ω) is a closed embedding i : X → M such that i∗ ω = 0.
Foundations of Geometry by Dixon E. T.