On the singularities of the exponential function of a semidirect product

We show that the Fréchet–Lie groups of the form $C^{\mathrm{\infty }}(M)⋊\mathbb{R}$ resulting from smooth flows on compact manifolds $M$ fail to be locally exponential in several cases: when at least one non-periodic orbit is locally closed, or when the flow restricts to a linear one on an orbit closure diffeomorphic to a torus. As an application, we prove that the Bondi–Metzner–Sachs group of symmetries of an asymptotically flat spacetime is not locally exponential.