Spectral Metric and Einstein Functionals

TL;DR

This paper introduces spectral functionals that produce metric and Einstein tensors on manifolds, extends these concepts to non-commutative geometry, and demonstrates the Einstein functional's vanishing in a specific non-commutative setting.

Contribution

It defines new bilinear functionals for geometric tensors and generalizes them to non-commutative spaces, including a key result for the noncommutative two-torus.

Findings

The functionals produce metric and Einstein tensors on manifolds.

Extension of these concepts to non-commutative geometry.

The Einstein functional vanishes for the conformally rescaled noncommutative two-torus.

Abstract

We define bilinear functionals of vector fields and differential forms, the densities of which yield the metric and Einstein tensors on even-dimensional Riemannian manifolds. We generalise these concepts in non-commutative geometry and, in particular, we prove that for the conformally rescaled geometry of the noncommutative two-torus the Einstein functional vanishes.