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32 LECTURE 2. COTANGENT BUNDLES 1. Since N X T ∗ X, we can find a neighborhood N0 of X in ∗ T X, a neighborhood N of X in M , and a diffeomorphism ψ : N0 → N such that ψ ✲ N ✛ ✲ N0 i i0 commutes . X Let ω0 = canonical form on T ∗ X ω1 = ψ ∗ ω symplectic forms on N0 . The submanifold X is lagrangian for both ω0 and ω1 . 2. There exist neighborhoods U0 and U1 of X in N0 and a diffeomorphism θ : U0 → U1 such that θ U0 ✲ U1 ✲ ✛ and θ ∗ ω1 = ω0 . i0 i0 commutes X Take ϕ = ψ ◦ θ and U = ϕ(U0 ). Check that ϕ∗ ω = θ∗ ψ ∗ ω = ω0 .

Taking S = {x} to be one point, the conormal bundle L = N ∗ S = is a cotangent fiber. 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, ·).

Proof. The graph Γϕ is lagrangian if and only if γ ∗ ω = 0. But γ∗ω = γ ∗ pr∗1 ω1 − γ ∗ pr∗2 ω2 = (pr1 ◦ γ)∗ ω1 − (pr2 ◦ γ)∗ ω2 and pr1 ◦ γ is the identity map on M1 whereas pr2 ◦ γ = ϕ. Therefore, γ∗ω = 0 ⇐⇒ ϕ∗ ω 2 = ω 1 . Lecture 3 Generating Functions Generating functions provide a method for producing symplectomorphisms via lagrangian submanifolds. We will illustrate their use in riemannian geometry and dynamics. We conclude with an application to the study of the group of symplectomorphisms and to the problem of the existence of fixed points, whose first instance is the Poincar´e-Birkhoff theorem.

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