  Landau levels play a key role in theoretical models of the quantum Hall
effect. Each Landau level is degenerate, flat and topologically non-trivial.
Motivated by Landau levels, we study tight-binding Hamiltonians whose energy
levels are all flat. We demonstrate that in two dimensions, for such
Hamiltonians, the flat bands must be topologically trivial. To that end, we
show that the projector onto each flat band is necessarily strictly local. Our
conclusions do not need the assumption of lattice translational invariance.
