Vortex motion on surfaces with constant curvature

Vortex motion on two-dimensional Riemannian surfaces with constant curvature is formulated. By way of the stereographic projection, the relation and difference between the vortex motion on a sphere (S-2) and on a hyperbolic plane (H-2) can be clearly analysed. The Hamiltonian formalism is presented for the motion of point vortices on S-2 and H-2. The set of first integrals for each Hamiltonian shows a corresponding algebraic property in terms of the Poisson bracket defined, respectively, for S-2 and H-2. As an example of analytic solutions, the motion of a vortex pair (dipole) is considered. It is shown that a dipole draws a geodesic curve as its trajectory on S-2 and H-2.