On the singularities of the exponential function of a semidirect product

We show that the Fréchet-Lie groups of the form C ∞ ( M ) ⋊ 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.