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lect06 - SYMPLECTIC GEOMETRY LECTURE 6 Prof Denis Auroux 1...

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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 6 Prof. Denis Auroux 1. Applications (1) The work done last time gives us a new way to look at T id Symp( M, ω ) (using C 1-topology, wherein f i : X Y converges to f iff f i f uniformly on compact sets and same for df i : T X T Y . → → → Now, f ∈ Symp( M, ω ) gives a graph graph( f ) = { ( x, f ( x )) } ⊂ ( M × M, pr ∗ 1 ω − pr 2 ∗ ω ) which is a Lagrangian submanifold. If f is C 1-close to the identity map, then graph( f ) is C 1-close to the diagonal Δ = { ( x, x ) } ⊂ ( M × M, pr ∗ 1 ω − pr 2 ∗ ω ) (i.e. the graph of the identity map). By Weinstein, a tubular neighborhood of Δ is diffeomorphic to U ⊂ ( T ∗ M, ω T ∗ M ), and the graph of f gives a section ( C 1-close to the zero section), i.e. the graph of a C 1-small µ ∈ Ω 1 ( M ). The fact that its graph is Lagrangian implies that µ is closed, i.e. dµ = 0. Thus, we have an identification T id (Symp( M, ω )) ∼ dµ = 0 } with = { µ ∈ Ω 1 | C 1 topologies.topologies....
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lect06 - SYMPLECTIC GEOMETRY LECTURE 6 Prof Denis Auroux 1...

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