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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 8 Prof. Denis Auroux 1. Almost-complex Structures Recall compatible triples ( , g, J ), wherein two of the three determine the third ( g ( u, v ) = ( u, Jv ) , ( u, v ) = g ( Ju, v ) , J ( u ) = g 1 ( ( u )) where g, are the induced isomorphisms T M T M ). Proposition 1. For ( M, ) a symplectic manifold with Riemannian metric g , a canonical almost complex structure J compatible with . 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 = ( , 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 -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, x ) of x-compatible complex structures on T x M is contractible, i.e. h t : J ( T x M, x ) J ( T x M, 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, ) is contractible. It is the space of sections of a bundle whose fibers are contractible by the previous proposition....
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