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Example text
The projection onto the configuration space of the two-torus is an annular region on S 2 . 6 Symplectic and Hamiltonian Actions Let (M, ω) be a symplectic manifold, and G a Lie group. Let ψ : G −→ Diff(M ) be a (smooth) action. , G acts by symplectomorphisms. In particular, symplectic actions of R on (M, ω) are in one-to-one correspondence with complete symplectic vector fields on M . Examples. 1. On R2n with ω = dxi ∧ dyi , let X = − ∂y∂ 1 . The orbits of the action generated by X are lines parallel to the y1 -axis, {(x1 , y1 − t, x2 , y2 , .
1 A riemannian metric on a manifold X is a function g which assigns to each point x ∈ X a positive inner product gx on Tx X. A riemannian metric g is smooth if for every smooth vector field v : X → T X the real-valued function x → gx (vx , vx ) is a smooth function on X. 2 A riemannian manifold (X, g) is a manifold X equipped with a smooth riemannian metric g. Let (X, g) be a riemannian manifold. The arc-length of a piecewise smooth curve γ : [a, b] → X is b arc-length of γ := a dγ dt , dt where dγ dt := gγ(t) dγ dγ , dt dt .
8 Weinstein Tubular Neighborhood Theorem Let (V, Ω) be a symplectic linear space, and let U be a lagrangian subspace. Claim. There is a canonical nondegenerate bilinear pairing Ω : V /U × U → R. Proof. Define Ω ([v], u) = Ω(v, u) where [v] is the equivalence class of v in V /U . Consequently, we get that Ω : V /U → U ∗ defined by Ω ([v]) = Ω ([v], ·) is an isomorphism, so that V /U U ∗ are canonically identified. In particular, if (M, ω) is a symplectic manifold, and X is a lagrangian submanifold, then Tx X is a lagrangian subspace of (Tx M, ωx ) for each x ∈ X.