Noncommutative geometry and lower dimensional volumes in Riemannian geometry

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 don't involve noncommutative geometry or spin structures at all.

12 pages

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• The Spectral Action Principle
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• Noncommutative Residue and sub-Dirac Operators for Foliations
• A Note on the Einstein-Hilbert action and the Dirac operator on R^n
• A Kastler-Kalau-Walze type theorem for the J-twist D_J of the Dirac operator
• Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature
• Sub-signature Operators and the Kastler-Kalau-Walze type theorem for manifolds with boundary
• The J-twist D_J of the Dirac operator and the Kastler-Kalau-Walze type theorem for six-dimensional manifolds with boundary
• Dirac-Witten Operators and the Kastler-Kalau-Walze type theorem for manifolds with boundary