Foliations and Diffeomorphism Groups — Question 12

Definitions and mathematical statement

A contact structure on a smooth three-dimensional manifold $M$ is a rank-two subbundle $\xi\subset TM$ that locally has the form $\xi=\ker\alpha$, where $\alpha$ is a smooth one-form satisfying $\alpha\wedge{\,\mathrm{d}}\alpha\neq 0$ at every point of its domain. A line field tangent to $\xi$ is a smooth rank-one subbundle $L\subset\xi$; this does not require a chosen orientation or a global nonzero section of $L$.

A smooth two-form $\omega\in\Omega^2(M)$ is closed and nowhere vanishing if \[ {\,\mathrm{d}}\omega=0,\qquad \omega_p\neq 0\quad\text{for every }p\in M. \] It annihilates $L$ if \[ \omega_p(v,w)=0\quad\text{for every }p\in M,\ v\in L_p,\ w\in T_pM. \] In dimension three, these conditions imply $\ker\omega_p=L_p$. Interpret "of integral class" as \[ [\omega]_{\mathrm{dR}}\in\operatorname{im}\left(H^2(M;{\mathbb{Z}})\longrightarrow H^2_{\mathrm{dR}}(M;{\mathbb{R}})\right), \] where the map is induced by change of coefficients and the de Rham isomorphism. The supplied record states no alternative normalization.

Question (Nariman--Yazdi, Question 12). How can one systematically construct examples of triples $((M,\xi),L,\omega)$ satisfying all the conditions above?

The request concerns a systematic production of examples; the source does not assert existence for every prescribed contact manifold or every prescribed tangent line field. It does not specify compactness, connectedness, boundary conditions, or coorientability of $\xi$, and it does not define what qualifies as a systematic construction. These qualifications remain unresolved rather than being added as hypotheses.

Short English statement

Find systematic constructions of contact three-manifolds equipped with a line field inside the contact planes and a nowhere-vanishing closed two-form of integral cohomology class whose kernel is that line field. The source leaves the intended scope of the examples and the meaning of "systematic" unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $800071$, exact source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 12." The supplied record attributes the question to Nariman and Yazdi, with proposed year 2025, and labels its status open; this status is supplied registry data, not an independently verified mathematical assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 12. Supplied source URL: https://arxiv.org/abs/2503.18274. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access or verification.