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

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.