Foliations and Diffeomorphism Groups — Question 10

Definitions and mathematical statement

In the standard smooth interpretation, an Engel manifold is a pair $(M,\mathcal D)$, where $M$ is a smooth four-dimensional manifold and $\mathcal D\subset TM$ is a smooth rank-two distribution satisfying \[ \mathcal D^2=\mathcal D+[\mathcal D,\mathcal D],\qquad \mathcal D^3=\mathcal D^2+[\mathcal D,\mathcal D^2],\qquad \operatorname{rank}(\mathcal D^2)=3,\quad \mathcal D^3=TM. \] Here brackets of distributions denote the pointwise spans of Lie brackets of their local smooth sections. Under this interpretation, define \[ \operatorname{Aut}(M,\mathcal D) =\{f\in\operatorname{Diff}^{\infty}(M): {\,\mathrm{d}} f_x(\mathcal D_x)=\mathcal D_{f(x)}\text{ for every }x\in M\}. \] The supplied Question 10, attributed there to a suggestion of Yoshi Mitsumatsu, asks whether a generic Engel manifold satisfies \[ \operatorname{Aut}(M,\mathcal D)=\{\operatorname{id}_M\}. \] The smooth category above is an explicit conventional interpretation, not a hypothesis specified by the supplied question. The record does not define "generic," specify a parameter space or topology, or state whether the underlying manifold is fixed. It also gives no compactness, connectedness, boundary, orientation, or boundary-preservation hypotheses. Consequently, the displayed equality specifies the requested rigidity property, but the supplied evidence does not determine a fully quantified genericity assertion.

Short English statement

Yoshi Mitsumatsu's suggested variation asks whether a generic Engel manifold has only the identity diffeomorphism preserving its Engel distribution. The meaning of "generic" and the class of manifolds and automorphisms under consideration require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $800069$, exact source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 10." The supplied record lists Nariman and Yazdi as proposers, gives the year 2025, and labels the problem open; this label is supplied curation, not an independently verified mathematical or literature assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 10. Supplied source URL: https://arxiv.org/abs/2503.18274. The supplied provenance states "Accessed: 2026-10-16" and extraction method arxiv_latex; these are supplied provenance statements, not a claim of independent access or verification here.
• The supplied question explicitly credits Yoshi Mitsumatsu with suggesting this variation of another question. The antecedent question is not included in the supplied record.