Abstract
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.
Similar content being viewed by others
References
Atiyah M., Bott R., Patodi V. (1973). On the heat equation and the index theorem. Invent. Math. 19: 279–330
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
Chamseddine A., Connes A. (1997). The spectral action principle. Commun. Math. Phys. 186(3): 731–750
Chamseddine, A., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing (E-print, arXiv, 2006)
Connes A. (1988). The action functional in noncommutative geometry. Commun. Math. Phys. 117(4): 673–683
Connes A. (1994). Noncommutative geometry. Academic, San Diego
Connes A. (1996). Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182(1): 155–176
Connes A., Moscovici H. (1995). The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2): 174–243
Dixmier J. (1966). Existence de traces non normales. C. R. Acad. Sci. Paris Sér.A-B 262: A1107–A1108
Getzler E. (1983). Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Commun. Math. Phys. 92: 163–178
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
Greiner P. (1971). An asymptotic expansion for the heat equation. Arch. Ration. Mech. Anal. 41: 163–218
Guillemin V. (1985). A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math. 55(2): 131–160
Guillemin V. (1993). Residue traces for certain algebras of Fourier integral operators. J. Funct. Anal. 115(2): 391–417
Kalau W., Walze M. (1995). Gravity, non-commutative geometry and the Wodzicki residue. J. Geom. Phys. 16(4): 327–344
Kastler D. (1995). The Dirac operator and gravitation. Commun. Math. Phys. 166(3): 633–643
Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. Progr. Math., vol. 131, pp. 173–197, Birkhäuser Boston, Boston, (1995)
Rennie, A., Varilly, J.: Reconstruction of manifolds in noncommutative geometry. E-print, arXiv, October 2006
Wodzicki M. (1984). Local invariants of spectral asymmetry. Invent. Math. 75(1): 143–177
Wodzicki, M.: Spectral asymmetry and noncommutative residue (in Russian), Habilitation Thesis, Steklov Institute, (former) Soviet Academy of Sciences, Moscow (1984)
Wodzicki, M.: Noncommutative residue. I. Fundamentals. Lecture Notes in Math., vol. 1289, pp. 320–399, Springer, Heidelberg (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-007-0199-2