Abstract
A 3-dimensional graph-manifold 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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REFERENCES
S. Buyalo and V. Kobel'skii, “Geometrization of graph-manifolds. I. Conformal geometrization,” Algebra Analiz, 7,No. 2, 3–45 (1995).
S. Buyalo and V. Kobel'skii, “Geometrization of graph-manifolds. II. Isometric geometrization,” Algebra Analiz, 7,No. 3, 96–117 (1995).
S. Buyalo and V. Kobel'skii, “Geometrization of infinite graph-manifolds,” Algebra Analiz, 8 (1996).
A. Chamseddine and A. Connes, “The spectral action principle,” Preprint IHES/M/96/37 (1996).
A. Connes, Non Commutative Geometry, Academic Press (1994).
A. Connes, “Non commutative geometry and physics,” Preprint IHES/M/93/32 (1993).
A. Connes, “Noncommutative geometry and reality,” J. Math. Phys., 36 (1995).
A. Connes, “Gravity coupled with matter and the foundation of noncommutative geometry,” Commun. Math. Phys., 182, 155–176 (1996).
A. Connes and H. Moscovici, “The local index formula in noncommutative geometry,” GAFA, 5, 174–243 (1995).
W. Kalau, “Hamilton formalism in noncommutative geometry,” J. Geom. Phys., 18, 349–380 (1996).
W. Kalau, N. A. Papadopoulos, J. Plass, and J.-M. Warzecha, “Differential algebras in noncommutative geometry,” J. Geom. Phys., 16, 149–167 (1995).
W. Kalau and M. Walze, “Gravity, noncommutative geometry, and the Wodzicki residue,” J. Geom. Phys., 16, 327–344 (1995).
T. Schücker and J.-M. Zylinski, “Connes' model building kit,” J. Geom. Phys., 16, 207–326 (1995).
A. Sitarz, “Noncommutative geometry and gauge theory on discrete groups,” J. Geom. Phys., 15, 123–136 (1995).
J. C. Várilly and J. M. Gracia-Bondia, “Connes' noncommutative differential geometry and the standard model,” J. Geom. Phys., 12, 223–301 (1993).
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Buyalo, S.V. Metrics of Nonpositive Curvature on Graph-Manifolds and Electromagnetic Fields on Graphs. Journal of Mathematical Sciences 119, 141–164 (2004). https://doi.org/10.1023/B:JOTH.0000008754.27256.5b
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DOI: https://doi.org/10.1023/B:JOTH.0000008754.27256.5b