Single-particle states in electronic Bloch bands form a Riemannian manifold whose geometric properties are described by two gauge invariant tensors, one being symmetric the other being antisymmetric, that can be combined into the so-called Fubini-Study metric tensor of the projective Hilbert space. The latter directly controls the Hall conductivity. Here we show that the symmetric part of the Fubini-Study metric tensor also has measurable consequences by demonstrating that it enters the current noise spectrum. In particular, we show that a non-vanishing equilibrium current noise spectrum at zero temperature is unavoidable whenever Wannier states have non-zero minimum spread, the latter being quantifiable by the symmetric part of the Fubini-Study metric tensor. We illustrate our results by three examples: (1) atomic layers of hexagonal boron nitride, (2) graphene, and (3) the surface states of three-dimensional topological insulators when gaped by magnetic dopants.