On the Classification of Flows on 2-Manifolds†

Publisher Summary This chapter presents the classification of flows on 2-manifolds. The set Σ of all structurally stable flows is open and dense in an approximation problem. The chapter presents a classification of all equivalent classes of Σ modulo Σ, thus, completing the solution of the fundamental problem. This is done by establishing a one-to-one correspondence between these equivalence classes and certain distinguished graphs, that is, graphs together with a distinguished set of edges satisfying some conditions. This results in a precise rule for labeling all equivalence classes of Σ modulo ˜ in such a way that each equivalence class appears exactly once in the labeling process. The chapter also discusses the concept of structurally stable flows, the graph of a gradient-like flow, the nonorientable case, without closed orbit, the orientable case with closed orbits, and the nonorientable case with closed orbits.