Foliations and Diffeomorphism Groups — Question 41

Definitions and mathematical statement

For a differentiable manifold $M$, let $\operatorname{Diff}^{1}(M)$ denote the group of bijections $f\colon M\to M$ for which both $f$ and $f^{-1}$ are continuously differentiable. The supplied record does not specify the hypotheses on $M$, including its dimension, compactness, or boundary conditions.

The symbols $\operatorname{CritCent}(f)$ and $\operatorname{CritReg}(f)$ are not defined in the supplied record. Their definitions, value domains, and the meaning of the comparison below must therefore be supplied before this can be a fully specified mathematical problem.

Subject to those unresolved qualifications, the source asks to characterize all $f\in\operatorname{Diff}^{1}(M)$ satisfying \[ \operatorname{CritCent}(f)>\operatorname{CritReg}(f), \] and to determine all possible values of $\operatorname{CritCent}(f)$ among diffeomorphisms satisfying this strict inequality. The record does not specify whether $M$ is fixed or also ranges over a class of manifolds.

Short English statement

Characterize the continuously differentiable diffeomorphisms whose source-designated quantity $\operatorname{CritCent}$ strictly exceeds $\operatorname{CritReg}$, and determine the possible values of $\operatorname{CritCent}$ in that case. The two quantities and the permitted manifolds are unspecified, so the formulation requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID $800100$, source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 41." The supplied record attributes the problem to Nariman and Yazdi, with proposed year 2025, and labels it open; this status is supplied metadata, not an independently verified mathematical assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 41. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance claims have not been independently verified here.