Fibers and global geometry of functions

TL;DR

This paper explores the use of fibers to analyze the global geometry of functions, offering new insights into the structure of folds and extending classical results with a geometric perspective.

Contribution

It introduces fiber-based methods to study global folds, providing a novel geometric framework that enhances understanding of function behavior.

Findings

Fibers offer a new perspective on the properties of folds.

Fiber-based descriptions generalize classical hypotheses.

The approach simplifies the analysis of global geometric structures.

Abstract

Since the seminal work of Ambrosetti and Prodi, the study of global folds was enriched by geometric concepts and extensions accomodating new examples. We present the advantages of considering fibers, a construction dating to Berger and Podolak's view of the original theorem. A description of folds in terms of properties of fibers gives new perspective to the usual hypotheses in the subject. The text is intended as a guide, outlining arguments and stating results which will be detailed elsewhere.