A NEW PARADIGM IN DIFFERENTIAL GEOMETRY: DEVELOPMENT, APPLICATIONS AND CURRENT CHALLENGES

Abstract

Differential Geometry, the discipline devoted to the rigorous analysis of the properties of differentiable manifolds and their associated smooth structures, has undergone a conceptual transformation in recent decades that has given rise to a new theoretical and applied paradigm. This approach incorporates methodologies from global analysis, differential topology, and mathematical physics, together with advanced tensor‐calculus techniques and geometric models in field theories and general relativity. Such convergence has expanded its sphere of influence to geometric computing, robotics, and high‐dimensional data modeling, demonstrating remarkable interdisciplinary potential. The recent evolution consolidates a renewed framework in which analytical, topological, algebraic, and computational approaches converge. The formulation of precise tools for studying curvatures, connections, and geometric flows facilitates the resolution of complex problems in general relativity, quantum field theory, and non‐Euclidean contexts. This paradigm examines current foundations, applications, and challenges, contributing to the expansion of its conceptual boundaries and its consolidation as a modeling instrument. Moreover, its integration into university teaching promotes the development of critical thinking and abstraction skills by combining formal rigor with dynamic visualizations and computational simulations, preparing future professionals to tackle contemporary scientific and technological challenges. This paradigm fosters interdisciplinary collaboration, consolidating its relevance in mathematical research.