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A note on asymptotic behaviors and topological properties of two smooth real-valued functions and several graphs associated to them

Naoki Kitazawa

Abstract

This is a note on the graphs of two smooth real-valued functions in the plane with no intersection and the natural map onto the region surrounded by them with the canonical projection to the line composed, yielding its Reeb space. The Reeb space of a real-valued function on a topological space is the set of all connected components of all level sets and topologized naturally. Such spaces have been fundamental and strong tools in theory of Morse functions and its generalization and variants, since the former half of the 20th century. They are graphs for tame functions such as Morse(-Bott) functions. The author has launched and has been studying this problem since 2020s, interested in Reeb spaces of smooth or non-analytic non-proper functions. For smooth closed manifolds and nice compact spaces, topological properties and combinatorial ones on Reeb spaces have been investigated by Gelbukh, Saeki, and so on.

A note on asymptotic behaviors and topological properties of two smooth real-valued functions and several graphs associated to them

Abstract

This is a note on the graphs of two smooth real-valued functions in the plane with no intersection and the natural map onto the region surrounded by them with the canonical projection to the line composed, yielding its Reeb space. The Reeb space of a real-valued function on a topological space is the set of all connected components of all level sets and topologized naturally. Such spaces have been fundamental and strong tools in theory of Morse functions and its generalization and variants, since the former half of the 20th century. They are graphs for tame functions such as Morse(-Bott) functions. The author has launched and has been studying this problem since 2020s, interested in Reeb spaces of smooth or non-analytic non-proper functions. For smooth closed manifolds and nice compact spaces, topological properties and combinatorial ones on Reeb spaces have been investigated by Gelbukh, Saeki, and so on.
Paper Structure (7 sections, 6 theorems)

This paper contains 7 sections, 6 theorems.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Differentiable manifolds and maps.
  4. (CW) complexes and graphs.
  5. Reeb spaces and Reeb (di)graphs.
  6. Our main result.
  7. Conflict of interest and Data availability.

Key Result

Proposition 1

Theorems & Definitions (9)

  • Proposition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • Theorem 4