Metrics of Nonpositive Curvature on GraphManifolds and Electromagnetic Fields on Graphs
 S. V. Buyalo
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A 3dimensional graphmanifold consists of simple blocks that are products of compact surfaces with boundary by the circle. Its global structure may be as complicated as desired and is described by a graph, which can be an arbitrary graph. A metric of nonpositive curvature on such a manifold, if it exists, can be described essentially by a finite number of parameters satisfying a geometrization equation. In the paper, it is shown that this equation is a discrete version of the Maxwell equations of classical electrodynamics, and its solutions, i.e., metrics of nonpositive curvature, are critical configurations of the same sort of action that describes the interaction of an electromagnetic field with a scalar charged field. This analogy is established in the framework of the spectral calculus (noncommutative geometry) of A. Connes. Bibliography: 17 titles.
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 Title
 Metrics of Nonpositive Curvature on GraphManifolds and Electromagnetic Fields on Graphs
 Journal

Journal of Mathematical Sciences
Volume 119, Issue 2 , pp 141164
 Cover Date
 20040101
 DOI
 10.1023/B:JOTH.0000008754.27256.5b
 Print ISSN
 10723374
 Online ISSN
 15738795
 Publisher
 Kluwer Academic PublishersPlenum Publishers
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 Authors

 S. V. Buyalo ^{(1)}
 Author Affiliations

 1. St.Petersburg Department of the Steklov Mathematical Institute, Russia