  In the last few decades, basic ideas of topology have completely transformed
the prediction of quantum transport phenomena. Following this trend, we go
deeper into the incorporation of modern mathematics into quantum material
science focusing on geometry. Here we investigate the relation between the
geometrical type of the Fermi surface and Anomalous and Spin Hall Effects. An
index, $\mathbb{H}_F$, quantifying the hyperbolic geometry of the Fermi
surface, shows a universal correlation (R$^2$ = 0.97) with the experimentally
measured intrinsic anomalous Hall conductivity, of 16 different compounds
spanning a wide variety of crystal, chemical, and electronic structure
families, including those where topological methods give R$^2$ = 0.52. This
raises a question about the predictive limits of topological physics and its
transformation into a wider study of bandstructures' and Fermi surfaces'
geometries, opening horizon for prediction of phenomena beyond topological
understanding.
