Contribution to the Study of Closed 1-Forms and Its Applications

One of the goals of this work is to emphasize the study with Professor Hounie, mainly along the papers (Hounie and Zugliani, Math Ann 369:1177–1209, 2017; Hounie and Zugliani, J Geom Anal 31:2540–2567, 2021), on closed 1-forms defined on manifolds. We also prove here a conjecture of V. I. Arnol’d by leveraging some results obtained during this collaboration, so we believe it is important to gather them, as additional details have been incorporated to some proofs. We tried to make the text as self-contained as possible for the interested reader. In light of the significance of the work done by Arnol’d, we hope it can be useful in other fields of Mathematics and Physics. Moreover, at the end, we present a novel application to the context we were initially interested in, namely, the solvability of certain involutive systems, and we also cast some open questions for further exploration in this domain.