On the first integral conjecture of Rene Thom

TL;DR

This paper proves Thom's conjecture that generic vector fields on compact manifolds have only trivial first integrals, extending the proof to locally Lipschitz integrals using a nonsmooth Sard theorem.

Contribution

It provides the first proof of Thom's conjecture for locally Lipschitz first integrals, broadening the class of functions for which the conjecture holds.

Findings

Thom's conjecture holds for locally Lipschitz first integrals.

A nonsmooth Sard theorem is used to establish the result.

The proof applies to generic vector fields on compact manifolds.

Abstract

More that half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals.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 \textquotedblleft generically\textquotedblright, 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 $C^{k}$ with $k\geq d,$ 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 $C^{1}$ sense.