The parameter rigid flows on oriented 3-manifolds

TL;DR

This paper characterizes parameter rigid flows on closed orientable 3-manifolds, proving they are smoothly conjugate to Kronecker flows on the 3-torus with badly approximable slopes, thus classifying their structure.

Contribution

It provides a complete classification of parameter rigid flows on closed orientable 3-manifolds as conjugate to specific Kronecker flows, a novel result in dynamical systems.

Findings

Parameter rigid flows on closed orientable 3-manifolds are classified.

Such flows are conjugate to Kronecker flows with badly approximable slopes.

The classification links flow rigidity to Diophantine approximation properties.

Abstract

A flow defined by a nonsingular smooth vector field $X$ on a closed manifold $M$ is said to be parameter rigid if given any real valued smooth function $f$ on $M$, there are a smooth funcion $g$ and a constant $c$ such that $f=X(g)+c$ holds. We show that the parameter rigid flows on closed orientable 3-manifolds are smoothly conjugate to Kronecker flows on the 3-torus with badly approximable slope.