Sobolev H1 geometry of the symplectomorphism group |
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Affiliation: | 1. School of Engineering and Advanced Technology, Massey University, Palmerston North, New Zealand;2. Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, Hamedan 65178, Iran |
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Abstract: | For a closed symplectic manifold with compatible Riemannian metric g we study the Sobolev geometry of the group of all diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the metric carries conjugate points via some simple examples. |
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Keywords: | Symplectic manifold Geodesic Hilbert manifold Fredholm operator Sobolev metric Conjugate point |
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