Noncommutative geometry and lower dimensional volumes in Riemannian geometry

TL;DR

This paper introduces a method to define lower-dimensional volumes, like area and length, of compact Riemannian manifolds using local invariants, inspired by noncommutative geometry but ultimately independent of it.

Contribution

It provides a new framework for defining and understanding lower-dimensional volumes in Riemannian geometry, connecting noncommutative geometry concepts with classical invariants.

Findings

Defined lower-dimensional volumes as integrals of local invariants

Connected noncommutative geometry with classical Riemannian invariants

Established definitions independent of noncommutative geometry and spin structures

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.