Sub-Riemannian-Driven Topological Restructuring of the Alpha Group Induced by the Angular Matrix M(θ)

This work investigates a mechanism for topology-altering geometric behavior induced by angular group actions within the framework of the Alpha Group, demonstrating that a systematic angular sweep can drive spontaneous transitions between Riemannian-like and sub-Riemannian regimes. Motivated by the group-theoretic foundations of geometry established by the Göttingen school and by the central role of invariants in topology, we introduce a setting in which geometric organization emerges from the interplay between an invariant, idempotent operator µ and an angular matrix M(θ ). While µ provides a stable structural background, the angular action associated with M(θ ) induces anisotropy and directional dependence. A systematic angular sweep is performed to probe the geometric response of the in- duced structure, revealing the spontaneous emergence of distinct isotropic, moderate, and strong regimes. These regimes are not imposed a priori, but arise dynamically from the interaction between the invariant background and the angular action. As the angular pa- rameter departs from stable configurations, anisotropic effects activate a sub-Riemannian Carnot–Carathéodory geometric framework, leading to a qualitative reorganization of the underlying topology. The strong regime is characterized by localized and persistent geometric features, ac- tivated only within bounded angular intervals and producing symmetric bifurcation pat- terns around θ = 90◦ . Despite these anisotropic reorganizations, global coherence and connectivity are preserved by the invariant action of µ. These results demonstrate that sub-Riemannian geometry can act as a natural driver of dynamic topological restructur- ing within group-based geometric frameworks, providing a coherent alternative to classical Riemannian descriptions.