  Geometry of quantum states has proved to be a useful concept for
understanding responses of electronic systems to static electromagnetic fields.
However, it has been challenging to relate quantum geometry with resonant
optical responses. The main obstacle is that optical transitions are properties
of a pair of states, while existing geometrical properties are defined for a
single state. Therefore, concrete geometric understanding of optical responses
has been limited to two-level systems, where one of two states determines the
Hilbert space completely. Here, we construct a general theory of Riemannian
geometry for resonant optical processes, by identifying transition dipole
moment matrix elements as tangent vectors. This theory applies to arbitrarily
high-order responses, suggesting that optical responses can be generally
thought of as manifestations of the Riemannian geometry of quantum states. We
use our theory to show that third-order photovoltaic Hall effects are related
to the Riemann curvature tensor and demonstrate an experimentally accessible
regime where they dominate the response.
