On the singularities of the exponential function of a semidirect product

TL;DR

This paper investigates the conditions under which certain infinite-dimensional Lie groups formed from smooth flows on manifolds fail to be locally exponential, with implications for the symmetry groups in general relativity.

Contribution

It demonstrates that specific classes of semidirect product Lie groups are not locally exponential, including those arising from particular flow dynamics and the Bondi--Metzner--Sachs group.

Findings

Certain flow-induced Lie groups are not locally exponential.

The Bondi--Metzner--Sachs group is not locally exponential.

Failure occurs when orbits are non-periodic and locally closed or when flows are linear on torus-like orbit closures.

Abstract

We show that the Fr\'echet--Lie groups of the form $C^{\infty}(M)\rtimes \mathbb{R}$ 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.