We show that a quadratic form of the quantum geometric tensor in $k$ space sets a bound on the $q^4$ term in the static structure factor $S\left(q\right)$ at small $\mathit{q}$. Bands that saturate this bound satisfy a condition similar to Laplace's equation, leading us to refer to them as harmonic bands. We provide examples of harmonic bands in one- and two-dimensional systems, including (higher) Landau levels. The geometric bound further leads to a topological bound on the $q^4$ term, which is saturated only when the band geometry satisfies the trace condition and, additionally, the quantum geometric tensor is uniform in $k$ space. We speculate that these bounds taken together provide a useful guide for identifying Chern bands that favor (Abelian or non-Abelian) fractional Chern insulators.

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• Chern insulators
• Electronic structure
• Geometric & topological phases
• Band structure methods