  We show that the topologically nontrivial bands of Chern insulators are
adiabatic cousins of the Landau bands of Hofstadter lattices. We demonstrate
adiabatic connection also between several familiar fractional quantum Hall
states on Hofstadter lattices and the fractional Chern insulator states in
partially filled Chern bands, which implies that they are in fact different
manifestations of the same phase. This adiabatic path provides a way of
generating many more fractional Chern insulator states and helps clarify that
nonuniformity in the distribution of the Berry curvature is responsible for
weakening or altogether destroying fractional topological states.
