Errata and Addenda to: "Hydrodynamic Vortex on Surfaces" and "The motion of a vortex on a closed surface of constant negative curvature"

TL;DR

This paper clarifies and completes the equations governing vortex motion on curved surfaces, confirming previous results and providing explicit formulas for the two-torus case, based on recent topological insights.

Contribution

It presents the corrected equations of Gustafsson for vortex dynamics on compact surfaces and verifies the validity of earlier conclusions despite initial equation incompleteness.

Findings

Previous results remain valid with corrected equations.

Explicit formulas for extra terms are derived for the two-torus.

The equations incorporate topological effects of the surface.

Abstract

The two papers in the title contain some equations that are not complete. The missing terms, which are of topological origin, were recently unveiled by Bjorn Gustafsson. In this note we present the equations of Gustafsson in the case of a single vortex in a compact boundaryless surface, and show that many conclusions we have reached with the incomplete equations remain valid. It seems that the the extra-terms in Gustafsson's equations can be explicitly written in terms of elementary functions only for the two-torus, and this is done at the end of this note.