  Motivated by the famous and pioneering mathematical works by Perelman,
Hamilton, and Thurston, we introduce the concept of using modern geometrical
mathematical classifications of multi-dimensional manifolds to characterize
electronic structures and predict non-trivial electron transport phenomena.
Here we develop the Fermi Surface Geometry Effect (FSGE), using the concepts of
tangent bundles and Gaussian curvature as an invariant. We develop an index,
$\mathbb{H}_F$, for describing the the "hyperbolicity" of the Fermi Surface
(FS) and show a universal correlation (R$^2$ = 0.97) with the experimentally
measured intrinsic anomalous Hall effect of 16 different compounds spanning a
wide variety of crystal, chemical, and electronic structure families, including
where current methods have struggled. This work lays the foundation for
developing a complete theory of geometrical understanding of electronic (and by
extension magnonic and phononic) structure manifolds, beginning with Fermi
surfaces. In analogy to the broad impact of topological physics, the concepts
begun here will have far reaching consequences and lead to a paradigm shift in
the understanding of electron transport, moving it to include geometrical
properties of the E vs k manifold as well as topological properties.
