A building block of non-commutative geometry is the observation that most of the geometric information of a compact Riemannian spin manifold M is encoded within its Dirac operator D. Especially via Connes' distance formula one is able to extract from the spectral properties of D the geodesic distance on M. In this paper we investigate the distance d encoded within a covariiant Dirac operator on a trivial U (n)-fiber bundle over the circle with arbitrary connection. It turns out that the connected components of d are tori whose dimension is given by the holonomy of the connection. For n = 2 we explicitly compute d on all the connected components. For n >= 2 we restrict to a given fiber and find that the distance is given by the trace of the module of a matrix. The latest is defined by the holonomy and the coordinate of the points under consideration. This paper extends to arbitrary n and arbitrary connection the results obtained in a previous work for U(2)-bundle with constant connection. It confirms interesting properties of the spectral distance with respect to another distance naturally associated to connection, namely the horizontal or Carnot-Caratheodory distance d(H). Especially in case the connection has irrational components, the connected components for d are the closure of the connected components of dH within the Euclidean topology on the torus. (C) 2008 Elsevier Inc. All rights reserved.
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|Titolo:||Spectral distance on the circle|
|Data di pubblicazione:||2008|
|Appare nelle tipologie:||01.01 - Articolo su rivista|