Nowhere-Zero Perturbations of Nonclosed One-Forms

Definitions and mathematical statement

Curated question. Let $M$ be a closed smooth manifold with $\chi(M)=0$, and let $\alpha\in\Omega^1(M;{\mathbb{R}})$ satisfy $d\alpha\ne0$. Does there exist $f\in C^\infty(M,{\mathbb{R}})$ such that\[\alpha_x+(df)_x\ne0\quad\text{in }T_x^*M\quad\text{for every }x\in M?\] Here closed conventionally means compact without boundary, $\chi$ is Euler characteristic, and $d$ is exterior differentiation. The perturbation is by an exact one-form; in particular $d(\alpha+df)=d\alpha$. The requested conclusion is pointwise nonvanishing of the covector, not closedness of the perturbed form.

The usual meaning of $d\alpha\ne0$ is that $d\alpha$ is not identically zero, so $\alpha$ is not closed. The stronger requirement $(d\alpha)_x\ne0$ at every point is not stated and must not silently replace it. Connectedness is also omitted: if disconnected manifolds are allowed, $\chi(M)=0$ only gives the sum of component Euler characteristics, whereas a nowhere-zero one-form requires vanishing Euler characteristic on each closed component. The intended connected or componentwise formulation must therefore be checked.

Incompatible source layers. The original statement field instead says: Extend the cylinder lemma for hyperbolic space forms to higher dimensions. No cylinder lemma, dimension range, or relation to the one-form implication is supplied. The verification field says the official report was checked and a higher-dimensional conjecture following the cylinder lemma was restored, but does not explain whether that conjecture is the original target or a separate adjacent problem. The original request and curated question are both retained, without asserting their equivalence.

The supplied triage checked 2026-08-21 labels the record open and discusses the higher-dimensional cylinder-lemma extension, reporting later skew-brane work as still describing its general conjecture as unproved. That assessment does not, from the supplied text, establish the status of the distinct one-form question. Both the source matching and the dated literature assessment require independent verification.

Short English statement

The curated question asks whether a nonclosed one-form on a closed Euler-characteristic-zero manifold can be made nowhere zero by adding an exact form. The original source field instead asks about a higher-dimensional cylinder lemma. Their relationship is missing, as are connectedness and the intended meaning of $d\alpha\ne0$.

Sources

• Local UnsolvedMath registry, permanent ID 30000136, code catalogue entry; curated title Nowhere-Zero Perturbations of Nonclosed One-Forms, Oberwolfach Reports Open Problems. The original field requests a higher-dimensional cylinder lemma for hyperbolic space forms.
• Geometrie (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/47, pp. 2491--2538. https://doi.org/10.4171/owr/2004/47. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label original report: https://doi.org/10.4171/owr/2004/47.
• Supplied literature label skew-brane paper: https://pure.mpg.de/rest/items/item_3123074_1/component/file_3123075/content.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports direct source checking and restoration of a conjecture following the cylinder lemma; this claim is not independently verified here.