Topological equivalence of smooth functions with isolated critical points on a closed surface

Abstract

We consider functions with isolated critical points on a closed surface. We prove that in a neighborhood of a critical point the function conjugates with Rez k for the some nonnegative integer k. The full topological invariant of such functions is constructed.1991 Mathematics Subject Classification. 57R45, 57R70, 58C27.

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

“…It is proved in [21] that if 0 ∈ R 2 = C is an isolated non-extremal critical point of a smooth function g: R 2 → R, then there is a homeomorphism h of C such that h(0) = 0 and g • h(z) = Re(z k ) for some k ≥ 1. It follows that the critical level-sets of a smooth map f : M → P having only isolated critical points are embedded graphs, i.e.…”

Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold

“…The function with isolated critical point, being not local extreme, on a closed surface is locally topologically equivalent with the following function: f (x, y) = Re(x + iy) k for some integer k, k ≥ 1 [15]. Therefore, firstly, we consider the level lines of function f (x, y) = Re(x+iy) k , defined on a surface R 2 + = {(x, y) ∈ R 2 | y ≥ 0} for k ∈ {1, 2, 3, 4} and in a general case.…”

Equivalence of closed 1-forms on closed nonorientable surfaces

“…It is known that if there exists a function f ∈ C 2 (G), where G is an open set, then ω = df if and only if ω is closed in G. For this reason, in what follows, we consider 1-forms ω for which there locally exists a function f : ω = df. It is known [5] that, for every critical point z 0 (except for local minima and maxima), there exists a neighborhood in which the function is conjugate to the function Re (x + iy) k for a certain number k ∈ N \{1}. Only two types of isolated points are possible, namely, a saddle and a center.…”