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Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group

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  • Let $M$ be a closed symplectic manifold with compatible symplectic form and Riemannian metric $g$. Here it is shown that the exponential mapping of the weak $L^{2}$ metric on the group of symplectic diffeomorphisms of $M$ is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the $L^{2}$ metric and Hofer's metric as well as an intriguing difference between the $L^{2}$ geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.
    Mathematics Subject Classification: Primary: 35Q05, 35Q83, 47H99, 58C99, 37K65; Secondary: 82D10.


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