# lect05 - SYMPLECTIC GEOMETRY LECTURE 5 Prof Denis Auroux...

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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 5 Prof. Denis Auroux Last time we proved: Theorem 1 (Moser) . ( M, ω 1 ) . Let M be a compact manifold, ( ω t ) symplectic forms, [ ω t ] constant = ⇒ ( M, ω ) ∼ = Theorem 2 (Darboux) . Locally, any symplectic manifold is locally isomorphic to ( R 2 n , ω ) . 1. Tubular Neighborhoods Let M n ⊃ X k be a submanifold with inclusion map i . Then we get a map d x i : T x X → T x M , with associated normal space N x X = T x M/T x X . Note that if there is a metric, one can identify this with the orthogonal space to X at x . Putting all these spaces together, we get a normal bundle NX = { ( x, v ) | x ∈ X, v ∈ N x X } with zero section i : X → NX, x → ( x, 0). Theorem 3. ∃ U a neighborhood of X in NX (via the-section) and U 1 a neighborhood of X in M s.t. ∃ φ : U ∼ U 1 a diffeomorphism. → Proof. (Idea) Equip M with a Riemannian metric g , so N x X ∼ T x X ⊥ ⊂ T x M . Then, given x ∈ X, v ∈ N x X → for | v | suﬃciently small ( | v | = g ( v, v ) < ), we obtain an exponential function exp x ( v ) (defined by considering a small geodesic segment with origin x and tangent vector v ). We obtain a map U → M, ( x, v ) → exp ( v ). For x x ∈ X, T ( x,...
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lect05 - SYMPLECTIC GEOMETRY LECTURE 5 Prof Denis Auroux...

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