We give an overview on the metric aspect of noncommutative geometry,especially the metric interpretation of gauge fields via the process of"fluctuation of the metric". Connes' distance formula associates to a gaugefield on a bundle P equipped with a connection H a metric. When the holonomy istrivial, this distance coincides with the horizontal distance defined by theconnection. When the holonomy is non trivial, the noncommutative distance hasrather surprising properties. Specifically we exhibit an elementary example ona 2-torus in which the noncommutative metric d is somehow more interesting thanthe horizontal one since d preserves the S^1-structure of the fiber and alsoguarantees the smoothness of the length function at the cut-locus. In thissense the fiber appears as an object "smoother than a circle". As aconsequence, from a intrinsic metric point of view developed here, any observerwhatever his position on the fiber can equally pretend to be "the center of theworld".
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