Integral symplectic pencils give compact fat torus bundles

Definitions and mathematical statement

For a principal $G$-bundle $P\to B$ with connection curvature $\Omega\in\Omega^2(B;\operatorname{Ad}P)$, use the standard Weinstein meaning of fatness: for every nonzero vertical Lie-algebra covector, the paired curvature is nondegenerate on the horizontal tangent space. With an invariant inner product this can be written using nonzero $u\in\mathfrak g$. The AIM question asks whether existence of a fat principal bundle forces $G\cong S^1, S^3$ or $SO(3)$. It does not explicitly require positive or nonnegative curvature of an entire connection-metric total space.

The supplied aid reports compact connected torus counterexamples under that literal definition. Let $B=T^4=({\mathbb{R}}/{\mathbb{Z}})^4$, $\mathrm{vol}={\,\mathrm{d}} x_1\wedge{\,\mathrm{d}} x_2\wedge{\,\mathrm{d}} x_3\wedge{\,\mathrm{d}} x_4$, and put \[ \omega_1={\,\mathrm{d}} x_1\wedge{\,\mathrm{d}} x_2+{\,\mathrm{d}} x_3\wedge{\,\mathrm{d}} x_4,\quad \omega_2={\,\mathrm{d}} x_1\wedge{\,\mathrm{d}} x_3-{\,\mathrm{d}} x_2\wedge{\,\mathrm{d}} x_4,\quad \omega_3={\,\mathrm{d}} x_1\wedge{\,\mathrm{d}} x_4+{\,\mathrm{d}} x_2\wedge{\,\mathrm{d}} x_3. \] These are closed integral forms satisfying \[ \omega_i\wedge\omega_j=2\delta_{ij}\,\mathrm{vol},\qquad \left(\sum_{i=1}^ru_i\omega_i\right)^2=2\sum_{i=1}^ru_i^2\,\mathrm{vol}\ne0 \quad(u\ne0), \] for $r=2,3$. Prequantize the first $r$ classes by principal circles ${\mathbb{R}}/(2\pi{\mathbb{Z}})$ with real connection curvature $2\pi\omega_i$, and take their fiber product. The resulting principal $T^r$-bundle has curvature $2\pi\sum_{i=1}^r\omega_i e_i$; every nonzero pairing is symplectic, so the connection is fat. Thus $T^2$ and $T^3$ disprove the stated group list. The same reasoning applies to integral rank-two subpencils; an arbitrary real subpencil still requires the period/lattice integrality needed for prequantization.

The aid also audits the flat-base connection metric: its abelian torus fibers have zero vertical sectional curvature, and a horizontal plane with nonzero curvature form has negative sectional curvature by O'Neill's formula. Hence this particular total-space metric is neither positively nor everywhere nonnegatively curved. Fatness governs the relevant mixed directions and does not imply positive curvature of every plane. If the workshop intended additional total-curvature hypotheses, these examples do not settle that stronger reading. The report refutes literal fatness alone and does not claim a classification with unspecified curvature assumptions.

Short English statement

Under standard fatness alone, the supplied integral symplectic pencil yields compact fat torus bundles, contradicting the three-group list. Their flat-base connection metrics have zero vertical and negative horizontal planes, so the examples do not answer a stronger question imposing total-space curvature.

Sources

• Problem 24, AIM workshop Manifolds with nonnegative sectional curvature, https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.
• The research summary identifies its fatness convention as Weinstein's, as used by Ziller and Florit--Ziller, and invokes O'Neill's curvature formula. Full bibliographic entries are not supplied.
• Local UnsolvedMath registry, permanent ID 20003149, exact code catalogue entry; canonical location aim-topology-notes.json, notes[236], dated 14 August 2026; statement labeled exact, registry status solved and research classification solved in literature.
• Research attempt 1: counterexample, problem refuted at the run, rigor label proved, novelty confidence low. The normalized torus construction and curvature audit are supplied machine-generated research aids, not peer reviewed; independent verification is required before citation.