Mathematical Physics, Analysis and Geometry

, Volume 15, Issue 4, pp 299–315 | Cite as

Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

  • Johannes Brunnemann
  • Christian FleischhackEmail author


Let \({\cal A}\) be the affine space of all connections in an SU(2) principal fibre bundle over ℝ3. The set of homogeneous isotropic connections forms a line l in \({\cal A}\). We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding \(l \hookrightarrow {\cal A}\) does not continuously extend to an embedding \(\overline{l} \hookrightarrow \overline{\cal A}\) of the respective compactifications. Here, the Bohr compactification \(\overline{l}\) corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and \(\overline{\cal A}\) to that of loop quantum gravity. Analogous results are given for the anisotropic case.


Parallel transports Spaces of connections Almost periodicity Qualitative theory of ODEs Loop quantum gravity Cosmological models 

Mathematics Subject Classifications (2010)

34C27 53C05 83F05 


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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany
  2. 2.Department MathematikUniversität HamburgHamburgGermany
  3. 3.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany

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