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Reeb spaces of functions being analytic on dense subsets and their graph structures

Naoki Kitazawa

Abstract

Reeb spaces of real-valued functions on manifolds are the spaces of all connected components (contours) of level sets and endowed with the natural quotient topology. They have been fundamental and strong tools in investigating manifolds via smooth functions with mild critical points since the birth of fundamental theory of Morse functions in the 20th century. We are concerned with topologies and combinatorics of them. Following an explicit note on explicit Reeb spaces of explicit functions which are real analytic (on dense sets) and seem to be simplest and most fundamental, edited by the author himself. We investigate other construction of examples of such functions and their Reeb spaces. Reeb spaces are naturally graphs in considerable cases and as another work, we also discuss natural definitions of vertices for them.

Reeb spaces of functions being analytic on dense subsets and their graph structures

Abstract

Reeb spaces of real-valued functions on manifolds are the spaces of all connected components (contours) of level sets and endowed with the natural quotient topology. They have been fundamental and strong tools in investigating manifolds via smooth functions with mild critical points since the birth of fundamental theory of Morse functions in the 20th century. We are concerned with topologies and combinatorics of them. Following an explicit note on explicit Reeb spaces of explicit functions which are real analytic (on dense sets) and seem to be simplest and most fundamental, edited by the author himself. We investigate other construction of examples of such functions and their Reeb spaces. Reeb spaces are naturally graphs in considerable cases and as another work, we also discuss natural definitions of vertices for them.
Paper Structure (4 sections, 3 theorems)

This paper contains 4 sections, 3 theorems.

Table of Contents

  1. Introduction.
  2. On Theorem \ref{['thm:1']}.
  3. On Theorem 2.
  4. Conflict of interest and Data availability.

Key Result

Theorem 1

There exist a connected infinite graph $G_0$ and an $m$-dimensional smooth connected submanifold $X_{m} \subset {\mathbb{R}}^{m+2}$ which is non-compact and represented as the zero set ${e_m}^{-1}(0)$ of some smooth map $e_m:{\mathbb{R}}^{m+2} \rightarrow {\mathbb{R}}^2$ with a function $c_{m}:X_{m}

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof : A proof of Theorem \ref{['thm:1']}
  • Theorem 3
  • proof : A proof of Theorem \ref{['thm:3']}, with Theorem \ref{['thm:2']}