Geometrical meaning of the Drude weight and its relationship to orbital magnetization

At the mean-field level the Drude weight is the Fermi-volume integral of the effective inverse mass tensor. I show here that the deviation of the inverse mass from its free-electron value is the real symmetric part of a geometrical tensor, which is naturally endowed with an imaginary antisymmetric part. The Fermi-volume integral of the latter yields the orbital magnetization. The novel geometrical tensor has a very compact form, and looks like a close relative of the familiar metric-curvature tensor. The Fermi-volume integral of each of the two tensors provides (via real and imaginary parts) a couple of macroscopic observables of the electronic ground-state. I discuss the whole quartet, for both insulating and metallic crystals.