Pseudo-Riemannian Metrics in Models Based on Noncommutative Geometry

TL;DR

This paper explores how metrics can be understood as elements of a tensor product in noncommutative geometry and generalizes the metric compatibility condition with linear connections within this framework.

Contribution

It introduces a novel perspective on metrics in noncommutative geometry, extending the compatibility condition to a left-linear tensor product setting.

Findings

Metrics as elements of left-linear tensor products of 1-forms

Generalization of metric compatibility condition

Framework applicable to models based on noncommutative differential calculus

Abstract

Several examples and models based on noncommutative differential calculi on commutative algebras indicate that a metric should be regarded as an element of the left-linear tensor product of the space of 1-forms with itself. We show how the metric compatibility condition with a linear connection generalizes to this framework.