Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group
James Benn
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.
keywords: conjugate point geodesic Fredholm map symplectic Euler equations. Diffeomorphism group Maxwell-Vlasov

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