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Geodesics On The Symplectomorphism Group

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Abstract

Let M be a compact manifold with a symplectic form ω and consider the group \({\mathcal{D}_\omega}\) consisting of diffeomorphisms that preserve ω. We introduce a Riemannian metric on M which is compatible with ω and use it to define an L 2-inner product on vector fields on M. Extending by right invariance we get a weak Riemannian metric on \({\mathcal{D}_\omega}\) . We show that this metric has geodesics which come from integral curves of a smooth vector field on the tangent bundle of \({\mathcal{D}_\omega}\) . Then, estimating the growth of such geodesics, we show that they extend globally.

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  1. Mathematics Department, Stony Brook University, Stony Brook, NY, 11794-3651, USA

    David G. Ebin

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  1. David G. Ebin
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Correspondence to David G. Ebin.

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In memory of Jerry Marsden

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Ebin, D.G. Geodesics On The Symplectomorphism Group. Geom. Funct. Anal. 22, 202–212 (2012). https://doi.org/10.1007/s00039-012-0150-2

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  • DOI: https://doi.org/10.1007/s00039-012-0150-2

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