Smooth Obstructions to Geometrically Semisimple Hamiltonians

Definitions and mathematical statement

The question is to identify smooth obstructions to the existence of a geometrically semisimple convex Hamiltonian. For a Hamiltonian $H$ on a symplectic manifold $(M,\omega)$, its vector field is defined, in one conventional sign choice, by $\iota_{X_H}\omega=-{\,\mathrm{d}} H$. In the common cotangent-bundle setting $M=T^*Q$, convexity usually refers to the restriction $H|_{T_q^*Q}$ on each cotangent fiber. The source does not specify that setting, strict convexity, superlinearity, or the differentiability class of $H$, so none of those extra requirements is imposed as a recovered hypothesis.

Geometric semisimplicity is a report-specific condition related to the geometry of an integrable system and its regular and singular parts. Its exact fibration, invariant-set, and tameness requirements are not reproduced. It must not be replaced by linear-algebraic semisimplicity or by an arbitrary weaker notion of integrability.

The requested obstructions concern which smooth underlying manifolds or phase spaces fail to admit a Hamiltonian satisfying both the intended convexity and geometric-semisimplicity conditions. The extraction does not identify that manifold class, say whether compactness is required, or specify whether the aim is necessary conditions, sufficient conditions, or a complete classification. It therefore supplies an obstruction problem rather than a stated equivalence. A precise formulation requires the exact domain, convexity assumptions, and definition of geometric semisimplicity from the report before any nonexistence criterion can be asserted.

Short English statement

Which features of the smooth manifold or phase space prevent a convex Hamiltonian from being geometrically semisimple? The exact phase space and geometric-semisimplicity definition are not supplied.

Sources

• Local UnsolvedMath registry, permanent ID $30001028$, source code catalogue entry, "Smooth Obstructions to Geometrically Semisimple Hamiltonians." No individual proposer is supplied.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406: https://doi.org/10.4171/owr/2008/42.
• Supplied reference, "Butler, Maslov cocycle and complete integrability": https://arxiv.org/abs/0708.3157.
• Supplied literature triage, checked 2026-08-21: status "open." It reports the smooth-obstruction question unanswered in the source paper and says partial obstructions and examples do not give a general classification. This is supplied dated curation, not independently verified; time-sensitive status and cited claims require independent verification.