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.