  Band geometry plays a substantial role in topological lattice models. The
Berry curvature, which resembles the effect of magnetic field in reciprocal
space, usually fluctuates throughout the Brillouin zone. Motivated by the
analogy with Landau levels, constant Berry curvature has been suggested as an
ideal condition for realizing fractional Chern insulators. Here we show that
while the Berry curvature cannot be made constant in a topological two-band
model, lattice models with three or more degrees of freedom per unit cell can
support exactly constant Berry curvature. However, contrary to the intuitive
expectation, we find that making the Berry curvature constant does not always
improve the properties of bosonic fractional Chern insulator states. In fact,
we show that an "ideal flatband" cannot have constant Berry curvature,
equivalently, we show that the density algebra of Landau levels cannot be
realised in any tight-binding lattice system.
