On the singularities of the exponential function of a semidirect product

TL;DR

The paper investigates the local exponentiality of Fréchet–Lie groups formed as semidirect products $C^ ablafty(M)\rtimes_
\mathbb{R}$ arising from smooth flows on compact manifolds $M$. By analyzing the exponential map through integral beta-operators $\beta_x$ and the spectral properties of infinitesimal generators, it derives explicit dynamical criteria that force non-local exponentiality, and applies these to prominent symmetry groups in general relativity. A central result is that the Bondi–Metzner–Sachs (BMS) group is not locally exponential, with analogous non-exponentiality established for conformal actions on spheres and related groups. The work also develops finite-differentiability refinements and Liouville-number phenomena to illuminate when exponential coordinates fail, informing the mathematical understanding of asymptotic symmetry groups in physics.

Abstract

We show that the Fréchet--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.

Theorems & Definitions (60)

• Theorem A
• Theorem B
• Conjecture A
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof