Beliebiger Eintrag,
A short guide through integration theorems of generalized distributions
(2017)cite arxiv:1710.01627Comment: 16 pages, v4: final version.
DOI: 10.1016/j.difgeo.2018.07.005
Zusammenfassung
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.
