Non-almost periodicity of parallel transports for homogeneous connections (Q1928787)

In homogeneous isotropic cosmological models, the classical configuration space is spanned by \(cA\) where \(c\in \mathbb{R}\) and \(A\) is a fixed homogeneous and isotropic connection. In fact, one does not consider these connections themselves but their parallel transports along certain edges. Usually, only straight edges have been taken into account and for such edges the parallel transports are periodic in the scalar \(c\), hence almost periodic. However, the concept of straightness requires a background metric. Therefore, it seems appropiate to consider general, non-straight edges. The main result of this paper proves that these edges, in general, do not lead to almost periodic parallel transports, whence they cannot be extended continuously to the Bohr compactification \(\overline{\mathbb{R}}_{Bohr}\). Technical arguments extend this result to the anisotropic case.