In the commutative case, a Hermitian line bundle with unitary connection corresponds to an essentially unique principal $\mathrm{U}(1)$-bundle with principal connection over the same base, which then admits a canonical lift of any given Riemannian metric on the base. Recent advances in noncommutatve Riemannian geometry permit a precise and cohesive generalization of this framework to the noncommutative setting through geometric analogues of Pimsner's construction. In this talk, I'll illustrate this generalization by sketching how complete Morita autoequivalences of a totally irrational noncommutative torus $C^\infty(\mathbb{T}^N_\theta)$ yield canonical lifts of differential, Riemannian, and metric geometry from $C^\infty(\mathbb{T}^N_\theta)$ to the corresponding noncommutative compact nilmanifolds—lifts that can involve distinct modular phenomena in vertical and horizontal directions.

This is partly based on joint work with T. Venkata Karthik.