Journal of Mathematical Sciences

, Volume 119, Issue 2, pp 141–164

Metrics of Nonpositive Curvature on Graph-Manifolds and Electromagnetic Fields on Graphs

  • S. V. Buyalo
Article

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

  1. 1.
    S. Buyalo and V. Kobel'skii, “Geometrization of graph-manifolds. I. Conformal geometrization,” Algebra Analiz, 7,No. 2, 3–45 (1995).Google Scholar
  2. 2.
    S. Buyalo and V. Kobel'skii, “Geometrization of graph-manifolds. II. Isometric geometrization,” Algebra Analiz, 7,No. 3, 96–117 (1995).Google Scholar
  3. 3.
    S. Buyalo and V. Kobel'skii, “Geometrization of infinite graph-manifolds,” Algebra Analiz, 8 (1996).Google Scholar
  4. 4.
    A. Chamseddine and A. Connes, “The spectral action principle,” Preprint IHES/M/96/37 (1996).Google Scholar
  5. 5.
    A. Connes, Non Commutative Geometry, Academic Press (1994).Google Scholar
  6. 6.
    A. Connes, “Non commutative geometry and physics,” Preprint IHES/M/93/32 (1993).Google Scholar
  7. 7.
    A. Connes, “Noncommutative geometry and reality,” J. Math. Phys., 36 (1995).Google Scholar
  8. 8.
    A. Connes, “Gravity coupled with matter and the foundation of noncommutative geometry,” Commun. Math. Phys., 182, 155–176 (1996).Google Scholar
  9. 9.
    A. Connes and H. Moscovici, “The local index formula in noncommutative geometry,” GAFA, 5, 174–243 (1995).Google Scholar
  10. 10.
    W. Kalau, “Hamilton formalism in noncommutative geometry,” J. Geom. Phys., 18, 349–380 (1996).Google Scholar
  11. 11.
    W. Kalau, N. A. Papadopoulos, J. Plass, and J.-M. Warzecha, “Differential algebras in noncommutative geometry,” J. Geom. Phys., 16, 149–167 (1995).Google Scholar
  12. 12.
    W. Kalau and M. Walze, “Gravity, noncommutative geometry, and the Wodzicki residue,” J. Geom. Phys., 16, 327–344 (1995).Google Scholar
  13. 13.
    T. Schücker and J.-M. Zylinski, “Connes' model building kit,” J. Geom. Phys., 16, 207–326 (1995).Google Scholar
  14. 14.
    A. Sitarz, “Noncommutative geometry and gauge theory on discrete groups,” J. Geom. Phys., 15, 123–136 (1995).Google Scholar
  15. 15.
    J. C. Várilly and J. M. Gracia-Bondia, “Connes' noncommutative differential geometry and the standard model,” J. Geom. Phys., 12, 223–301 (1993).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • S. V. Buyalo
    • 1
  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteRussia

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