The noncommutative residue, divergence theorems and the spectral geometry functional

TL;DR

This paper introduces a novel proof of the divergence theorem using noncommutative residues and extends it to manifolds with boundary, also defining and calculating spectral divergence functionals.

Contribution

It provides a new proof of the divergence theorem via noncommutative residues and extends it to manifolds with boundary, introducing spectral divergence functionals.

Findings

New proof of divergence theorem using Dirac operator and noncommutative residues

Extension of divergence theorem to manifolds with boundary via noncommutative residue of B-algebra

Calculation of spectral divergence functionals for manifolds with or without boundary

Abstract

In this paper, a simple proof of the divergence theorem is given by using the Dirac operator and noncommutative residues. Then we extend the divergence theorem to compact manifolds with boundary by the noncommutative residue of the B-algebra. Furthermore, we present some spectral geometric functionals which we call spectral divergence functionals, and we calculate the spectral divergence functionals of manifolds with (or without) boundary.