Conjugate points on the symplectomorphism group

Abstract

Let $\mathcal {D}_{\omega }^{s}(M)$ denote the group of symplectic diffeomorphisms of a closed symplectic manifold $M$, which are of Sobolev class $H^{s}$ for sufficiently high $s$. When equipped with the $L^{2}$ metric on vector fields, $\mathcal {D}_{\omega }^{s}$ becomes an infinite-dimensional Hilbert manifold whose tangent space at a point $\eta$ consists of $H^{s}$ sections $X$ of the pull-back bundle $\eta ^{*}TM$ for which the corresponding vector field $u=X\circ \eta ^{-1}$ on $M$ satisfies $\mathcal {L}_{u}\omega =0$. Geodesics of the $L^{2}$ metric are globally defined, so that the $L^{2}$ metric admits an exponential mapping defined on the whole tangent space. It was shown that this exponential mapping is a non-linear Fredholm map of index zero. Singularities of the exponential map are known as conjugate points and in this paper we construct explicit examples of them on $\mathcal {D}_{\omega }^{s}(\mathbb {C}P^{n})$. We then solve the Jacobi equation explicitly along a geodesic in $\mathcal {D}_{\omega }^{s}$, generated by a Killing vector field, and characterize all conjugate points along such a geodesic.