Conjugate points on the symplectomorphism group (Q494734)

The author studies the conjugate points on the group of all symplectomorphisms of the Sobolev class \(H^s\) with sufficient large \(s\). By equipped with the \(L^2\) metric, the geodesics are defined globally and an exponential mapping is defined on the whole tangent space. Then the author shows that this exponent mapping is a non-linear Fredholm map of index zero, and the singularities of the exponent map are known as conjugate points. Finally, the author characterizes all conjugate points along a geodesic by solving the Jacobi equation explicitly along such a geodesic in a group generated by a Killing vector field.