Characterize non-local exponentiality for semidirect products C^∞(M) ⋊ R from flows

Characterize completely, for smooth flows σ: R × M → M on compact manifolds M, when the associated Fréchet–Lie groups C^∞(M) ⋊_α R fail to be locally exponential by identifying necessary and sufficient conditions on σ under which the exponential map of C^∞(M) ⋊_α R is not a local diffeomorphism at the identity.

Background

The authors develop several sufficient criteria that imply non-local exponentiality of C^∞(M) ⋊ R, and they apply these to important examples, including the BMS group. However, they emphasize that these results do not yet provide a full characterization of when non-local exponentiality occurs across all flows.

This statement indicates that a definitive, necessary-and-sufficient classification of non-local exponentiality for C^∞(M) ⋊ R remains unresolved.