Ideal Chern bands as Landau levels in curved space

We prove that all the criteria proposed in the literature to identify a Chern band hosting exact fractional Chern insulating ground states, in fact, describe an equivalence with a lowest Landau level defined in curved space under a nonuniform magnetic field. In addition, we design an operational test for the most general instance of such lowest Landau-level mapping, which only relies on the computationally inexpensive evaluation of Bloch wavefunctions' derivatives. Our work clarifies the common origin of various Chern-idealness criteria, proves that these criteria exhaust all possible lowest Landau levels, and hints at classes of Chern bands that may possess interesting phases beyond Landaulevel physics.