  We demonstrate that flat bands with local Berry curvature arise naturally in
chiral (ABC) multilayer graphene placed on a boron nitride (BN) substrate. The
degree of flatness can be tuned by varying the number of graphene layers N. For
N = 7 the bands become nearly flat, with a small bandwidth of 3.6 meV. The two
nearly flat bands coming from the K and K' valleys cross along lines in the
reduced zone. Weak intervalley tunneling turns the bandcrossing into an avoided
crossing, producing two nearly flat bands with global Chern number zero, but
with local Berry curvature. The flatness of the bands suggests that many body
effects will dominate the physics, while the local Berry curvature of the bands
endows the system with a nontrivial quantum geometry. The quantum geometry
effects manifest themselves through the quantum distance (Fubini-Study) metric,
rather than the more conventional Chern number. Multilayer graphene on BN thus
provides a platform for investigating the effect of interactions in a system
with a non-trivial quantum distance metric, without the complication of
non-zero Chern numbers. We note in passing that flat bands with non-zero Chern
number can also be realized by making use of magnetic adatoms, and explicitly
breaking time reversal symmetry.
