Almost everything in mathematics, indeed, can be "hyperbolic", "parabolic" or "elliptic". Like PDE's, Riemann surfaces, or manifolds of higher dimension, fractional-linear transformations, fixed points of a map, etc.
Not even mentioning the 3 kinds of the conic sections:-)
Of course this can be traced back to the fundamental trichotomy "positive", "zero" and "negative".
In differential geometry we have three great areas: "positive curvature", "negative curvature" and "zero curvature".
