Geometric Characterization of Liouville Integrability via a Curvature Atlas for Rigid-Body Dynamics

Abstract: We introduce a curvature atlas for left-invariant metrics on SU(2), based on the inertial curvature field derived from the Euler-Poincare equations. We prove that the classical integrable cases of the heavy top--spherical, Lagrange, Kovalevskaya, and Goryachev-Chaplygin--correspond precisely to degenerate curvature signatures of this field, namely isotropic, orthogonally split, and symmetric-pair signatures. This yields a geometric necessary and sufficient condition for Liouville integrability: the geodesic flow (and the heavy top with axis-symmetric potential) is integrable if and only if the curvature signature is degenerate. Beyond the classical list, the atlas reveals a balanced-mixed regime (inertia ratio 2:2:1) that, while non-integrable, admits an exact curvature-balance relation and a family of pure-precession solutions. We formulate a curvature deviation functional quantifying the distance to integrability, describe near-integrable dynamics near the 2:2:1 regime, and present a complete integrability map in the plane of inertia ratios. The work provides a unified geometric framework for classifying, perturbing, and controlling rigid-body systems.