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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.03956 |
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| _version_ | 1866910010977550336 |
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| author | Simić, Slobodan N. |
| author_facet | Simić, Slobodan N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03956 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence of global cross sections to volume-preserving flows Simić, Slobodan N. Dynamical Systems 37C10, 58A10 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. |
| title | On the existence of global cross sections to volume-preserving flows |
| topic | Dynamical Systems 37C10, 58A10 |
| url | https://arxiv.org/abs/2602.03956 |