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

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.

• Publication year

  2026

• Venue

  Unknown venue

• Publication date

  2026-02-03

• Fields of study

  Mathematics

• Identifiers
  arXiv 2602.03956
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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