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

S. V. Buyalo

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 28-72

Résumé

A 3-dimensional graph-manifold $M$ consists of simple blocks, which are products of compact surfaces with boundary by the circle. The global structure $M$ can be as complicated as ane likes and is described by a graph which can be arbitrary. A metric of nonpositive curvature (an NPC-metric) on $M$, if it exists, is described essentially by a finite number of parameters satisfying a geometrization equation. In the paper, this equation is shown to be a discrete version of the Maxwell equations of classical electrodynamics, and its solutions, i.e., NPC-metrics on $M$, are critical configurations of the same sort of action that describes interaction of an electromagnetic field with a scalar charged field. This analogy is established in the framework of A. Connes' spectral calculs (noncommutative geometry).