Closed-one-form twists obstruct the asserted cyclic-cover product

Definitions and mathematical statement

The extracted Burago--Kleiner Conjecture 7.3.1 asserts that the metric on the fixed ${\mathbb{Z}}$-cover associated with the $S^1$-factor of $M\times S^1$ is a direct product \[ (M\times{\mathbb{R}},\widetilde g)\simeq(M,g_1)\times({\mathbb{R}},{\,\mathrm{d}} t^2). \] The aid interprets the missing preceding setting as a metric without conjugate points on a closed product manifold. The actual preceding hypotheses are not quoted in the supplied record, so the counterexample scope must remain the literal fixed-cover interpretation, rather than an assertion about every possible fuller formulation.

For that interpretation, let $(\Sigma,h)$ be a closed connected oriented hyperbolic surface and let $\alpha$ be a smooth closed one-form with $[\alpha]\ne0\in H^1_{\mathrm{dR}}(\Sigma)$. On $\Sigma\times({\mathbb{R}}/{\mathbb{Z}})$, with ${\,\mathrm{d}}\theta$ the global angular form, put \[ g_\alpha=h+({\,\mathrm{d}}\theta+\alpha)^2. \] Locally $\alpha={\,\mathrm{d}} f$, so the coordinate $u=\theta+f$ makes this a local product. Hence its sectional curvatures are nonpositive and it has no conjugate points. The fixed cyclic cover defined by projection to ${\mathbb{R}}/{\mathbb{Z}}$, equivalently by the kernel of the corresponding homomorphism $\pi_1(\Sigma\times S^1)\to{\mathbb{Z}}$, has metric \[ \widetilde g_\alpha=h+({\,\mathrm{d}} t+\alpha)^2\quad\text{on }\Sigma\times{\mathbb{R}}. \] The field $W=\partial_t$ is unit and parallel, with dual one-form $\eta={\,\mathrm{d}} t+\alpha$. Its periods are the period character \[ \pi_1(\Sigma)\longrightarrow{\mathbb{R}},\qquad [c]\longmapsto\int_c\alpha. \] They are nonzero for some loop, so $\eta$ is not exact.

On the universal cover, a primitive of the lifted $\alpha$ gives the ordinary product $\widetilde\Sigma\times{\mathbb{R}}$. Its hyperbolic factor has no parallel fields, so the only parallel direction is $W$. If the specified cyclic cover were isometric to $(K,k)\times{\mathbb{R}}$ with a compact surface factor $K$, the unit line field of that product would pull back to $\pm W$, whose dual would be exact. This contradicts the period character. Thus the fixed cover is not such a product, even under an unmarked isometry. In this family the exact criterion is \[ (\Sigma\times{\mathbb{R}},h+({\,\mathrm{d}} t+\alpha)^2) \text{ is a compact-factor product}\quad\Longleftrightarrow\quad[\alpha]=0. \] For the reverse implication, $\alpha={\,\mathrm{d}} f$ and $u=t+f$ give the global product isometry. A product structure on the universal cover is fully consistent with this failure on the specified cyclic cover.

More generally, in a trivialization $B\times{\mathbb{R}}$ adapted to a free proper parallel unit translation field $\partial_t$, the invariant metric has the form \[ h_B+({\,\mathrm{d}} t+A)^2,\qquad {\,\mathrm{d}} A=0. \] A product trivialization preserving that line field and the base exists exactly when $[A]=0$. To promote this marked criterion to an obstruction against every unmarked product decomposition, one needs uniqueness of the parallel line, as in the hyperbolic example; if additional parallel directions exist, nonexactness of the chosen dual form alone does not suffice.

The source also suggests broader questions for no-conjugate-points metrics on products and manifolds admitting higher-rank nonpositively curved metrics, without stating them precisely. Its page heading and the start of Section 7.4 about extending conjugate-point notions to magnetic geodesics and natural Hamiltonian systems are adjoining context, not additional conjectures to reconstruct.

Short English statement

A closed nonexact one-form twists a locally product metric on a surface times a circle. The universal cover splits, but the cyclic cover fixed by the original circle projection has a parallel direction with nonzero periods and cannot split with a compact factor. The omitted source hypotheses still need checking.

Sources

• Permanent UnsolvedMath ID $20001908$, exact public code catalogue entry; local registry aim-geometry-notes.json, notes[245], canonical tag conjecture.
• AIM workshop "Geodesics," Geometry, Section 7.3, Conjecture 7.3.1, attributed in text to Burago--Kleiner: https://aimath.org/pastworkshops/geodesicsproblems.pdf. No PDF was accessed for this draft. Original literature, proposer/year metadata and full bibliographic citations are absent.
• Research attempt 1, "Closed-one-form twists obstruct the asserted cyclic-cover product," labelled counterexample, refuted at its run, proved and low novelty confidence; registry solved and SOLVED-IN-LITERATURE. The supplied verification note says preceding-record hypotheses were lost at the boundary but does not quote them in this record. The exact field preserves the extracted statement, not a recovered full hypothesis list.