  We show that the dipole moment of an exciton is uniquely determined by the
quantum geometry of its eigenstates, and demonstrate its intimate connection
with a quantity we call the dipole curvature. The dipole curvature arises
naturally in semiclassical dynamics of an exciton in an electric field, adding
a term additional to the anomalous velocity coming from the Berry's curvature.
In a uniform electric field this contributes a drift velocity akin to that
expected for excitons in crossed electric and magnetic fields, even in the
absence of a real magnetic field. We compute the quantities relevant to
semiclassical exciton dynamics for several interesting examples of bilayer
systems with weak interlayer tunneling and Fermi energy in a gap, where the
exciton may be sensibly described as a two-body problem. These quantities
include the exciton dispersion, its dipole curvature, and Berry's curvature.
For two gapped-graphene layers in a vanishing magnetic field, we find the
dipole curvature vanishes if the layers are identical, but may be non-zero when
the layers differ. We further analyze examples in the presence of magnetic
fields, allowing us to examine cases involving graphene, in which a gap is
opened by Landau level splitting. Heterostructures involving TMDs are also
considered. In each case the dipole and/or Berry's curvatures play out
differently. In some cases, the lowest energy exciton state is found to reside
at finite momentum, with interesting possibilities for Bose condensation. We
also find situations in which the dipole curvature increases monotonically with
exciton momentum, suggesting that the quantum geometry can be exploited to
produce photocurrents from initially bound excitons with electric fields. We
speculate on further possible effects of the semiclassical dynamics in
geometries where the constituent layers are subject to the same or different
electric fields.
