کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4592709 1335130 2008 38 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Spectral distance on the circle
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Spectral distance on the circle
چکیده انگلیسی

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 covariant 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–Carathéodory distance dH. 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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Functional Analysis - Volume 255, Issue 7, 1 October 2008, Pages 1575-1612