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Noncommutative Geometry and Lower Dimensional Volumes in Riemannian Geometry


In this paper we explain how to define “lower dimensional” volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes do not involve noncommutative geometry or spin structures at all.

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  1. 1.

    Atiyah M., Bott R., Patodi V. (1973). On the heat equation and the index theorem. Invent. Math. 19: 279–330

  2. 2.

    Branson T., Gilkey P.B., Ørsted B. (1990). Leading terms in the heat invariants for the Laplacians of the de Rham, signature, and spin complexes. Math. Scand. 66: 307–319

  3. 3.

    Chamseddine A., Connes A. (1997). The spectral action principle. Commun. Math. Phys. 186(3): 731–750

  4. 4.

    Chamseddine, A., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing (E-print, arXiv, 2006)

  5. 5.

    Connes A. (1988). The action functional in noncommutative geometry. Commun. Math. Phys. 117(4): 673–683

  6. 6.

    Connes A. (1994). Noncommutative geometry. Academic, San Diego

  7. 7.

    Connes A. (1996). Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182(1): 155–176

  8. 8.

    Connes A., Moscovici H. (1995). The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2): 174–243

  9. 9.

    Dixmier J. (1966). Existence de traces non normales. C. R. Acad. Sci. Paris Sér.A-B 262: A1107–A1108

  10. 10.

    Getzler E. (1983). Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Commun. Math. Phys. 92: 163–178

  11. 11.

    Gilkey P.B. (1995). Invariance theory, the heat equation and the Atiyah-Singer index theorem, 2nd edn. Studies in Advanced Mathematics. CRC Press, Boca Raton

  12. 12.

    Greiner P. (1971). An asymptotic expansion for the heat equation. Arch. Ration. Mech. Anal. 41: 163–218

  13. 13.

    Guillemin V. (1985). A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math. 55(2): 131–160

  14. 14.

    Guillemin V. (1993). Residue traces for certain algebras of Fourier integral operators. J. Funct. Anal. 115(2): 391–417

  15. 15.

    Kalau W., Walze M. (1995). Gravity, non-commutative geometry and the Wodzicki residue. J. Geom. Phys. 16(4): 327–344

  16. 16.

    Kastler D. (1995). The Dirac operator and gravitation. Commun. Math. Phys. 166(3): 633–643

  17. 17.

    Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. Progr. Math., vol. 131, pp. 173–197, Birkhäuser Boston, Boston, (1995)

  18. 18.

    Rennie, A., Varilly, J.: Reconstruction of manifolds in noncommutative geometry. E-print, arXiv, October 2006

  19. 19.

    Wodzicki M. (1984). Local invariants of spectral asymmetry. Invent. Math. 75(1): 143–177

  20. 20.

    Wodzicki, M.: Spectral asymmetry and noncommutative residue (in Russian), Habilitation Thesis, Steklov Institute, (former) Soviet Academy of Sciences, Moscow (1984)

  21. 21.

    Wodzicki, M.: Noncommutative residue. I. Fundamentals. Lecture Notes in Math., vol. 1289, pp. 320–399, Springer, Heidelberg (1987)

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Correspondence to Raphaël Ponge.

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Ponge, R. Noncommutative Geometry and Lower Dimensional Volumes in Riemannian Geometry. Lett Math Phys 83, 19–32 (2008). https://doi.org/10.1007/s11005-007-0199-2

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Mathematics Subject Classification (2000)

  • Primary 58J42
  • Secondary 53B20
  • 58J40


  • Noncommutative geometry
  • local Riemannian geometry
  • pseudodifferential operators