Fredholm properties of the \(L^2\) exponential map on the symplectomorphism group (Q254834)

The paper showed 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, where \(M\) is a closed symplectic manifold with compatible symplectic form and Riemannian metric \(g\). The result provides an 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 diffeomorphisms group.