Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections

TL;DR

This paper constructs metrized quantum vector bundles over quantum tori using Riemannian metrics and Rosenberg's Levi-Civita connections, and shows that scalar multiples of a metric are at zero distance in the modular Gromov-Hausdorff propinquity.

Contribution

It introduces a method to build metrized quantum vector bundles from Riemannian metrics on quantum tori and analyzes their metric equivalence under scalar multiplication.

Findings

Scalar multiples of a Riemannian metric have zero distance in the modular Gromov-Hausdorff propinquity.

Construction of metrized quantum vector bundles from Riemannian metrics on quantum tori.

Extension of Rosenberg's Levi-Civita connections to the setting of quantum tori.

Abstract

We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero between them with respect to the modular Gromov-Hausdorff propinquity.