Cross sections to flows via intrinsically harmonic forms

TL;DR

This paper introduces a new criterion for the existence of a global cross section in volume-preserving flows on compact manifolds, based on the intrinsic harmonicity of a specific differential form.

Contribution

It establishes a novel equivalence between the existence of a global cross section and the intrinsic harmonicity of a certain form in the context of volume-preserving flows.

Findings

Global cross section exists iff the (n-1)-form is intrinsically harmonic.

Provides a new geometric criterion for analyzing flow dynamics.

Connects harmonic forms with flow cross sections in differential geometry.

Abstract

We establish a new criterion for the existence of a global cross section to a non-singular volume-preserving flow on a compact manifold. Namely, if $\Phi$ is a non-singular smooth flow on a compact, connected manifold $M$ with a smooth invariant volume form $\Omega$, then $\Phi$ admits a global cross section if and only if the $(n-1)$-form $i_X \Omega$ is intrinsically harmonic, that is, harmonic with respect to some Riemannian metric on $M$.