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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 14/1997

S. V. Buyalo
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Metrics of nonpositive curvature
on graph-manifolds
and electromagnetic fields on graphs

This preprint was accepted August, 1997.

Contact: ` S. V. Buyalo `

Abstract:

A 3-dimensional graph-manifold is composed from simple blocks
which are products of compact surfaces with boundary by the
circle. Its global structure may be as complicated as one likes and is
described by a graph which might 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 which satisfy a
geometrization equation. The aim of the work is to show 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
which describes the interaction of an electromagnetic field with a
scalar charged field. To establish this analogy it turns out to be
possible in the framework of the spectral calculus (noncommutative
geometry) of A. Connes.

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