Nonpositive curvature on the area-preserving diffeomorphism group

Elementary consequences of the criterion are also discussed: for example, there are no flows with nonpositive curvature operator on the standard round sphere; and on a flat surface, every rotationally symmetric flow has nonpositive curvature operator. Finally we show that if a steady flow satisfies both this nonpositive curvature criterion and the well-known Eulerian stability criterion of Arnold, then all Lagrangian perturbations grow polynomially in time, in the L2 norm. Thus this is the first time methods of Riemannian geometry have given rigorous information on stability.