  Quantum geometry is a fundamental concept to characterize the local
properties of quantum states. It is recently demonstrated that saturating
certain quantum geometric bounds allows a topological Chern band to share many
essential features with the lowest Landau level, facilitating fractionalized
phases in moir\'e flat bands. In this work, we systematically extend the
consequence and universality of saturated geometric bounds to arbitrary Landau
levels by introducing a set of single-particle states, which we term as
``generalized Landau levels''. These generalized Landau levels exhibit exactly
quantized values of integrated trace of quantum metric determined by their
corresponding Landau level indices, regardless of the nonuniformity of their
quantum geometric quantities. We derive all geometric quantities for individual
and multiple generalized Landau levels, discuss their relations, and understand
them in light of the theory of holomorphic curves and moving frames. We further
propose a model by superposing few generalized Landau levels which is supposed
to capture a large portion of the single-particle Hilbert space of a generic
Chern band analogous to the first Landau level. Using this model, we employ
exact diagonalization to identify a single-particle geometric criterion for
permitting the non-Abelian Moore-Read phase, which is potentially useful for
future engineering of moir\'e materials and beyond. We use a double twisted
bilayer graphene model with only adjacent layer hopping term to show the
existence of first generalized Landau level type narrow band and zero-field
Moore-Read state at the second magic angle which serves as a promising starting
point for more detailed future studies. We expect that generalized Landau
levels will serve as a systematic tool for analyzing topological Chern bands
and fractionalized phases therein.
