On the existence of global cross sections to volume-preserving flows

TL;DR

This paper provides a new criterion involving Riemannian metrics and volume forms for the existence of a global cross section in volume-preserving flows on closed manifolds.

Contribution

It introduces a novel condition based on the codifferential and Riemannian metrics that guarantees the existence of a global cross section for volume-preserving flows.

Findings

A new criterion involving the codifferential for global cross sections.

Existence of a Riemannian metric making the canonical form harmonic.

Conditions under which flows admit global cross sections.

Abstract

We establish a new criterion for the existence of a global cross section to a non-singular volume-preserving flow $\Phi$ on a closed smooth manifold $M$. Namely, if $X$ is the infinitesimal generator of the flow and $\Phi$ preserves a smooth volume form $\Omega$, then $\Phi$ admits a global cross section if there exists a smooth Riemannian metric $g$ on $M$ with Riemannian volume $\Omega$ and $g(X,X) = 1$ such that $\lVert \delta_g (i_X \Omega) \rVert_g < 1$, where $\delta_g$ denotes the codifferential relative to $g$; (equivalently, $\lVert dX^\flat \rVert_g < 1$). In that case, there in fact exists another smooth Riemannian metric on $M$ with respect to which the canonical form $i_X \Omega$ is co-closed and therefore harmonic.