Metrics of nonpositive curvature on graph-manifolds and electromagnetic fields on graphs

TL;DR

This paper explores the analogy between nonpositive curvature metrics on graph-manifolds and electromagnetic fields, using noncommutative geometry to connect geometrical structures with classical electrodynamics equations.

Contribution

It demonstrates that the geometrization equation for nonpositive curvature on graph-manifolds is a discrete analogue of Maxwell's equations, linking geometry with electromagnetic theory.

Findings

The geometrization equation is a discrete version of Maxwell's equations.

Metrics of nonpositive curvature correspond to critical points of an electromagnetic action.

The framework uses noncommutative geometry to establish the analogy.

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. We establish this analogy in the framework of the spectral calculus (noncommutative geometry) of A. Connes.