A short guide through integration theorems of generalized distributions

TL;DR

This paper clarifies the various formulations and historical development of the Stefan-Sussmann theorem on the integrability of generalized distributions, highlighting errors and influences in geometric control theory from 1962 to 1994.

Contribution

It provides a detailed analysis of the different statements, proofs, and historical context of the Stefan-Sussmann theorem, correcting misconceptions and tracing its evolution.

Findings

Identifies and corrects errors in previous formulations of the theorem.

Clarifies the influence of Hermann and other mathematicians on the development.

Provides a comprehensive genealogy of ideas related to integrability of generalized distributions.

Abstract

The generalization of Frobenius' theorem to foliations with singularities is usually attributed to Stefan and Sussmann, for their simultaneous discovery around 1973. However, their result is often referred to without caring much on the precise statement, as some sort of magic spell. This may be explained by the fact that the literature is not consensual on a unique formulation of the theorem, and because the history of the research leading to this result has been flawed by many claims that turned to be refuted some years later. This, together with the difficulty of doing proof-reading on this topic, brought much confusion about the precise statement of Stefan-Sussmann's theorem. This paper is dedicated to bring some light on this subject, by investigating the different statements and arguments that were put forward in geometric control theory between 1962 and 1994 regarding the problem of integrability of generalized distributions. We will present the genealogy of the main ideas and show that many mathematicians that were involved in this field made some mistakes that were successfully refuted. Moreover, we want to address the prominent influence of Hermann on this topic, as well as the fact that some statements of Stefan and Sussmann turned to be wrong. In this paper, we intend to provide the reader with a deeper understanding of the problem of integrability of generalized distributions, and to reduce the confusion surrounding these difficult questions.