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Unformatted text preview: • SYMPLECTIC GEOMETRY, LECTURE 8 Prof. Denis Auroux 1. Almost-complex Structures Recall compatible triples ( ω, g, J ), wherein two of the three determine the third ( g ( u, v ) = ω ( u, Jv ) , ω ( u, v ) = g ( Ju, v ) , J ( u ) = ˜ g − 1 (˜ ω ( u )) where ˜ g, ω ˜ are the induced isomorphisms T M T ∗ M ). → Proposition 1. For ( M, ω ) a symplectic manifold with Riemannian metric g , ∃ a canonical almost complex structure J compatible with ω . Idea. Do polar decomposition on every tangent space. Corollary 1. Any symplectic manifold has compatible almost-complex structures, and the space of such struc- tures is path connected. Proof. For the first part, using a partition of unity gives a Riemannian metric, so the rest follows from the proposition. For the second part, given J , J 1 , let g i = ω ( , J i · ) for i = 0 , 1 and set g t = (1 − t ) g + tg 1 . Each · of these (for t ∈ [0 , 1]) is a metric, and gives an ω-compatible J ˜ t by polar decomposition, with J ˜ = J and J ˜ 1 = J 1 . The mechanism of the proof also gives Proposition 2. The set J ( T x M, ω x ) of ω x-compatible complex structures on T x M is contractible, i.e. ∃ h t : J ( T x M, ω x ) → J ( T x M, ω x ) for t ∈ [0 , 1] , h = id , h 1 = J → J , h t ( J ) = J ∀ t . Corollary 2. The space of compatible almost-complex structures on ( M, ω ) is contractible. It is the space of sections of a bundle whose fibers are contractible by the previous proposition....
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- Spring '10
- Geometry, Symplectic manifold, Vector bundle, Prof. Denis Auroux, almost-complex structures, compatible almost-complex structures