Non-Almost Periodicity of Parallel Transports for Homogeneous Connections |
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Authors: | Johannes Brunnemann Christian Fleischhack |
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Institution: | 1. Institut f??r Mathematik, Universit?t Paderborn, 33095, Paderborn, Germany 2. Department Mathematik, Universit?t Hamburg, Bundesstra?e 55, 20146, Hamburg, Germany 3. Max-Planck-Institut f??r Mathematik in den Naturwissenschaften, Inselstra?e 22?C26, 04103, Leipzig, Germany
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Abstract: | Let ${\cal A}$ be the affine space of all connections in an SU(2) principal fibre bundle over ?3. 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. |
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