  We consider twisted bilayer graphene on a transition metal dichalcogenide
substrate, where proximity-induced spin-orbit coupling significantly alters the
eight flat bands which occur near the magic angle. The resulting band structure
features a pair of extremely flat bands across most of the mini-Brillouin zone.
Further details depend sensitively on the symmetries of the heterostructure; we
find semiconducting band structures when all two-fold rotations around in-plane
axis are broken, and semi-metallic band structures otherwise. We calculate the
Chern numbers of the different isolated bands, and identify the parameter
regimes and filling factors where valley Chern insulators and topological
insulators are realized. Interestingly, we find that for realistic values of
the proximity-induced terms, it is possible to realize a topological insulator
protected by time-reversal symmetry by doping two holes or two electrons per
superlattice unit cell into the system.
