Reeb spaces of smooth functions on manifolds II

TL;DR

This paper characterizes smooth functions on closed manifolds whose Reeb spaces form finite graphs and provides explicit examples illustrating various properties of these functions and their Reeb spaces.

Contribution

It offers a characterization of smooth functions with Reeb spaces as finite graphs and presents explicit examples with interesting properties.

Findings

Reeb spaces of certain smooth functions are finite graphs.

Explicit examples of functions with special Reeb space properties.

Characterization criteria for functions with graph-structured Reeb spaces.

Abstract

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth functions on closed manifolds such that they themselves or their Reeb spaces have some interesting properties.