Reeb spaces of smooth functions associated to globally similar graphs of smooth functions

TL;DR

This paper explores the topological and combinatorial properties of Reeb spaces of smooth functions with globally similar graphs, extending previous work on functions with converging or diverging graphs at infinity.

Contribution

It introduces a new perspective on Reeb spaces for functions with globally similar graphs, expanding the understanding of their topological structure and properties.

Findings

Reeb spaces can be characterized as graphs under certain conditions.

The study extends previous work on functions with specific asymptotic behaviors.

Connections to non-proper functions and their Reeb spaces are discussed.

Abstract

Previously, we have investigated a natural smooth map onto the region surrounded by the graphs of two smooth real-valued functions in the plane converging to a same value or diverges to $+\infty$ or $-\infty$ simultaneously, at each infinity, and topological properties and combinatorial ones of its composition with the canonical projection. Here, we consider smooth functions with congruent or globally similar graphs instead. Here, the Reeb space of a smooth function on a manifold with no boundary is fundamental and important. This is the naturally topologized quotient space of the manifold, consisting of all connected components (contours) of the function and is a graph under a certain nice situation. Related studies also related to the present study were started due to interest of the author in theory of Reeb spaces of non-proper functions. For proper functions, in 2020s related studies have developed mainly due to Gelbukh and Saeki.