Noncommutative geometry and lower dimensional volumes in Riemannian geometry

Details Noncommutative geometry and lower dimensional volumes in Riemannian geometry Noncommutative geometry and lower dimensional volumes in Riemannian geometry

2007-07-28

arxiv.org/abs/0707.4201v1

Lett. Math. Phys. 83 (2008) 19-32

Mathematics

Differential Geometry

12 pages

Scientific paper

10.1007/s11005-007-0199-2

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.

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