Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on Reeb spaces of some explicit real analytic functions

Naoki Kitazawa

TL;DR

The paper investigates realizing Reeb spaces of real-analytic maps, with a focus on infinite graphs, by constructing explicit real-analytic manifolds $X$ and maps $c:X\\to\\mathbb{R}$ that are obtained as restrictions of canonical projections ${\\pi}_{n,1}$. Using a regional, real-algebraic construction, it ensures properness of the quotient map and prescribed topology of level sets, including regular contours diffeomorphic to $S^{m-1}$. The main results include Theorem 1, which yields a non-compact real-analytic $X$ and a map $c$ with $R_c$ isomorphic to a connected infinite graph $G$, and Theorem 2, which produces a compact real-analytic $X$ with $R_c$ homeomorphic to a Peano continuum $G$ whose punctured version is an $E$-graph; the construction also yields an induced $E$-graph structure on appropriate subspaces. These findings extend Reeb-space realizations beyond finite graphs and illuminate the structure of $R_c$ in the real-analytic category, providing explicit constructive techniques via projections and algebraic regions.

Abstract

Reeb spaces of smooth functions are fundamental and strong tools in understanding manifolds via smooth functions with mild critical points. They are defined as the natural spaces of all connected components of level sets. They are also important objects in related studies. Realization of graphs as Reeb spaces of smooth functions of certain nice classes is of such studies. In this paper, we present Reeb spaces of explicit real analytic functions which are not finite graphs. Related problems were started by Sharko, in 2006, who has studied smooth functions with critical points represented by certain elementary polynomials, and followed by a study of Masumoto and Saeki, which is on smooth functions on closed surfaces under an extended situation, and a study of Michalak, which is on Morse functions on closed manifolds. The author has contributed to this by respecting topologies of level sets, and real algebraic construction.

A note on Reeb spaces of some explicit real analytic functions

TL;DR

The paper investigates realizing Reeb spaces of real-analytic maps, with a focus on infinite graphs, by constructing explicit real-analytic manifolds and maps that are obtained as restrictions of canonical projections . Using a regional, real-algebraic construction, it ensures properness of the quotient map and prescribed topology of level sets, including regular contours diffeomorphic to . The main results include Theorem 1, which yields a non-compact real-analytic and a map with isomorphic to a connected infinite graph , and Theorem 2, which produces a compact real-analytic with homeomorphic to a Peano continuum whose punctured version is an -graph; the construction also yields an induced -graph structure on appropriate subspaces. These findings extend Reeb-space realizations beyond finite graphs and illuminate the structure of in the real-analytic category, providing explicit constructive techniques via projections and algebraic regions.

Abstract

Reeb spaces of smooth functions are fundamental and strong tools in understanding manifolds via smooth functions with mild critical points. They are defined as the natural spaces of all connected components of level sets. They are also important objects in related studies. Realization of graphs as Reeb spaces of smooth functions of certain nice classes is of such studies. In this paper, we present Reeb spaces of explicit real analytic functions which are not finite graphs. Related problems were started by Sharko, in 2006, who has studied smooth functions with critical points represented by certain elementary polynomials, and followed by a study of Masumoto and Saeki, which is on smooth functions on closed surfaces under an extended situation, and a study of Michalak, which is on Morse functions on closed manifolds. The author has contributed to this by respecting topologies of level sets, and real algebraic construction.
Paper Structure (3 sections, 2 theorems)

This paper contains 3 sections, 2 theorems.

Table of Contents

  1. Introduction.
  2. A proof of Theorems \ref{['thm:1']} and \ref{['thm:2']}.
  3. Conflict of interest and Data availability.

Key Result

Theorem 1

There exists a connected infinite graph $G$ and for each integer $m>1$, we have an $m$-dimensional real analytic manifold $X \subset {\mathbb{R}}^{m+1}$ which is non-compact and represented as the zero set of a real analytic function and a function $c:X \rightarrow \mathbb{R}$ with the following pro

Theorems & Definitions (4)

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