  We propose a new kind of geometric effective theory based on curved
space-time single valley Dirac theory with spin connection for twisted bilayer
graphene under generic twist angle. This model can reproduce the nearly flat
bands with particle-hole symmetry around the first magic angle. The band width
is near the former results given by Bistritzer-MacDonald model or density
matrix renormalization group. Even more, such geometric formalism allows one to
predict the properties of rotating bilayer graphene which cannot be accessed by
former theories. As an example, we investigate the Bott index of a rotating
bilayer graphene. We relate this to the two-dimensional Thouless pump with
quantized charge pumping during one driving period which could be verified by
transport measurement.
