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

Naoki Kitazawa

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.

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

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 or 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.
Paper Structure (5 sections, 4 theorems)

This paper contains 5 sections, 4 theorems.

Table of Contents

  1. Introduction.
  2. Our main result.
  3. Main Theorems.
  4. Future problems.
  5. Conflict of interest and Data availability.

Key Result

Theorem 1

Let $m \geq 2$ be an integer. Let $c_i:\mathbb{R} \rightarrow \mathbb{R}$ ($i=1,2$ ) be smooth functions satisfying $c_1(x)<c_2(x)$ for any $x \in \mathbb{R}$. Let $D_{c_1,c_2}:=\{(x_1,x_2)\mid c_1(x_2)<x_1<c_2(x_2)\}$ and we use $\overline{D_{c_1,c_2}}$ for its closure in ${\mathbb{R}}^2$. Then $X_

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • proof
  • Proposition 1
  • Proposition 2
  • proof