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Fundamental Examples of non-compact Reeb spaces of smooth functions

Naoki Kitazawa

Abstract

Reeb spaces of (continuous) real-valued functions on (nice) topological spaces are the spaces whose underlying sets consist of all connected components (contours) of their level sets and seen naturally as quotient spaces of the spaces. They are "$1$-dimensional" spaces in various nice cases. They are graphs or graphs with ends for smooth function cases with nice singularities and behaviors. These tools or objects, have been fundamental and important in theory of Morse functions and more general smooth functions and applications to geometry, since the 20th century. We present new systematic methods to obtain Reeb spaces homeomorphic to infinite graphs (with ends) for functions on non-compact manifolds with no boundary. This note is regarded as a note on cases previously obtained by the author from a kind of general viewpoints.

Fundamental Examples of non-compact Reeb spaces of smooth functions

Abstract

Reeb spaces of (continuous) real-valued functions on (nice) topological spaces are the spaces whose underlying sets consist of all connected components (contours) of their level sets and seen naturally as quotient spaces of the spaces. They are "-dimensional" spaces in various nice cases. They are graphs or graphs with ends for smooth function cases with nice singularities and behaviors. These tools or objects, have been fundamental and important in theory of Morse functions and more general smooth functions and applications to geometry, since the 20th century. We present new systematic methods to obtain Reeb spaces homeomorphic to infinite graphs (with ends) for functions on non-compact manifolds with no boundary. This note is regarded as a note on cases previously obtained by the author from a kind of general viewpoints.
Paper Structure (3 sections, 3 theorems)

This paper contains 3 sections, 3 theorems.

Table of Contents

  1. Introduction.
  2. Our main result.
  3. 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 such that $c_1(x)<c_2(x)$ for any $x \in \mathbb{R}$. Then consider a region $D_{c_1,c_2}:=\{(x_1,x_2)\mid c_1(x_2)<x_1<c_2(x_2)\}$ and its closure $\overline{D_{c_1,c_2}}$ in ${\mathbb{R}}^2$. T

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof