We prove that all the criteria proposed in the literature to identify Chern bands hosting fractional Chern insulating ground states, in fact, describe an equivalence with lowest Landau levels defined in curved space under a non-uniform magnetic field. In addition, we design an operational test for the most general instance of 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 posses interesting phases beyond Landau level physics.