On the first integral conjecture of René Thom

Jacky Cresson, Aris Daniilidis, Masahiro Shiota

Research output: Contribution to journalArticleResearchpeer-review

Abstract

More than half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word "generically", this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (few) known formal proofs of Thom's conjecture are all relying to the classical Sard theorem and are thus requiring the technical assumption that first integrals should be of class Ck with k ≥ d, where d is the dimension of the manifold. In this work, using a recent nonsmooth extension of Sard theorem we establish the validity of Thom's conjecture for locally Lipschitz first integrals, interpreting genericity in the C1 sense. © 2007 Elsevier Masson SAS. All rights reserved.
Original languageEnglish
Pages (from-to)625-631
JournalBulletin des Sciences Mathematiques
Volume132
DOIs
Publication statusPublished - 1 Oct 2008

Keywords

  • First integral
  • Sard theorem
  • Structural stability
  • o-minimal structure

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