Vortices on closed surfaces

Understand

We consider $N$ point vortices $s_j$ of strengths $\kappa_j$ moving on a closed (compact, boundaryless, orientable) surface $S$ with riemannian metric $g$.

• As far as we know, only the sphere or surfaces of revolution, the latter qualitatively, have been treated in the available literature.
• The aim of this note is to present an intrinsic geometric formulation for the general case.
• We give a simple proof of Kimura's conjecture that a dipole describes geodesic motion.