  We describe a few elementary aspects of the circle of ideas associated with a
quantum field theory (QFT) approach to Riemannian Geometry, a theme related to
how Riemannian structures are generated out of the spectrum of (random or
quantum) fluctuations around a background fiducial geometry. In such a
scenario, Ricci curvature with its subtle connections to diffusion, optimal
transport, Wasserestein geometry, and renormalization group, features
prominently.
