Geodesics On The Symplectomorphism Group

Let M be a compact manifold with a symplectic form ω and consider the group D_w{\mathcal{D}_\omega} consisting of diffeomorphisms that preserve ω. We introduce a Riemannian metric on M which is compatible with ω and use it to define an L ²-inner product on vector fields on M. Extending by right invariance we get a weak Riemannian metric on D_w{\mathcal{D}_\omega} . We show that this metric has geodesics which come from integral curves of a smooth vector field on the tangent bundle of D_w{\mathcal{D}_\omega} . Then, estimating the growth of such geodesics, we show that they extend globally.