Abstract: In condensed matter, the problem of electrons hopping on a two-dimensional lattice naturally leads to the notion of a Chern band. Mathematically, a Chern band is described by a smooth map from the Brillouin zone—a torus labeling the unitary characters of the Abelian group of lattice translations—to a complex projective space. The geometry of these maps plays an important role in shaping physical responses of quantum materials and effective interactions, especially when the band dispersion is flat. In particular, the so-called ideal bands are quite remarkable from the physical point of view as they reproduce the physics of the lowest energy subspace—the lowest Landau level—for electrons in a uniform magnetic field. Ideal Kähler bands correspond to the case where the map is holomorphic. Recently, we have introduced the notion of generalized Landau levels (GLLs), which provide lattice analogs of higher Landau levels. Mathematically, GLLs correspond to Chern bands for which the associated map from the Brillouin zone to complex projective space is harmonic. In this talk we will review the theory, and discuss some recent results on the rigidity of such maps, in the same spirit of Calabi's rigidity theorem for isometric embeddings of complex manifolds in complex projective space.