Toward the classification of cohomology-free vector fields

TL;DR

This paper proves Katok's conjecture for 3-manifolds, showing that only tori support cohomology-free vector fields and these are smoothly conjugate to Diophantine ones, advancing understanding of dynamical systems on manifolds.

Contribution

The paper provides a proof of Katok's conjecture specifically for 3-manifolds, confirming the classification of cohomology-free vector fields in this case.

Findings

Cohomology-free vector fields only exist on tori in 3-manifolds

Such vector fields are smoothly conjugate to Diophantine vector fields

The proof confirms the conjecture in the 3-dimensional case

Abstract

In 1984, Anatole Katok conjectured that the only closed orientable manifolds that support cohomology-free vector fields are tori and these vector fields are smoothly conjugated to Diophantine (constant) ones. In this work we present a proof of Katok conjecture for 3-manifolds.