On the existence of global cross sections to volume-preserving flows
Slobodan N. Simić
TL;DR
The paper provides a new criterion for the existence of a global cross section for a non-singular volume-preserving flow $\Phi$ on a closed manifold $M$. It shows that if there is a Riemannian metric $g$ with volume form $\Omega$ satisfying $\lVert \delta_g (i_X \Omega)\rVert_g < m_g(X)^2$ (where $X$ is the flow's generator and $m_g(X)=\inf_p \lVert X\rVert_g$), then $\Phi$ admits a global cross section and $M$ fibers over $S^1$; moreover, $i_X \Omega$ can be made co-closed (harmonic) for some metric. The proof leverages a distance-to-closed-forms estimate, constructs a nearby closed form, shows a transverse integrable plane field, and invokes intrinsic harmonicity results to deduce the cross section, with Plante-type consequences for the fibration structure. The work bridges dynamical properties of flows with Hodge-theoretic conditions and provides a practical criterion to certify global cross sections in volume-preserving dynamics. This has implications for reducing flow dynamics to first-return maps and clarifies when geodesic-type flows (e.g., generic unit tangent bundle flows) lack global cross sections due to harmonic constraints.
Abstract
We establish a new criterion for the existence of a global cross section to a non-singular volume-preserving flow $Φ$ on a closed smooth manifold $M$. Namely, if $X$ is the infinitesimal generator of the flow and $Φ$ preserves a smooth volume form $Ω$, then $Φ$ admits a global cross section if there exists a smooth Riemannian metric $g$ on $M$ with Riemannian volume $Ω$ and $g(X,X) = 1$ such that $\lVert δ_g (i_X Ω) \rVert_g < 1$, where $δ_g$ 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 $i_X Ω$ is co-closed and therefore harmonic.