Conjecture A: Non-local exponentiality of C^∞(M) ⋊ R for every non-trivial flow

Prove that for every smooth non-trivial flow σ: R × M → M on a compact manifold M, the Fréchet–Lie group C^∞(M,R) ⋊_α R, where α_t(f) = f ∘ σ_t, is not locally exponential; more specifically, show that there exists a sequence t_n > 0 with t_n → 0 such that the points (0, t_n) are singular for the exponential map of C^∞(M,R) ⋊_α R.

Background

The paper studies semidirect product Lie groups of the form C^∞(M) ⋊ R arising from smooth flows σ: R × M → M on compact manifolds. It provides several sufficient criteria under which these groups fail to be locally exponential (e.g., presence of a non-periodic locally closed orbit, linear flow on a torus, or existence of an equivariant submersion onto S^1 or a torus).

Motivated by these results and applications such as the BMS group in general relativity, the authors formulate a general conjecture asserting that non-local exponentiality holds for all non-trivial flows, with a precise prediction about sequences approaching 0 that yield singular points of the exponential map.