Differential Geometry: A Unified Constructive Approach to Exterior Differential Equations and Geometric PDEs

This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to exterior differential geometric equations on smooth manifolds. We define the exterior differential geometric closure KExtDiffGeo, a differentially closed field extension constructed through recursive adjunction processes that incorporate geometric objects, harmonic forms, curvature tensors, and fundamental solutions for exterior differential operators.Within this closure, we prove that solutions to broad classes of exterior differential geometric equations—including Yang-Mills equations, Einstein-Cartan theory, and complex exterior equations on Kähler manifolds—admit unified representations that respect the underlying geometric and algebraic structures. The framework rigorously addresses nonlinearity, geometric constraints, and multi-dimensional challenges while preserving graded algebraic structures and exterior algebraic compatibility conditions. We provide detailed constructive proofs, derive explicit solution formulas with rigorous error bounds, and establish convergence criteria in appropriate geometric function spaces. Comprehensive algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with certified error bounds. A rigorous validation framework employing interval arithmetic and discrete exterior calculus demonstrates the practical effectiveness of our approach. The work demonstrates that explicit analytic solutions exist within appropriately constructed differential algebraic closures, providing new algebraic perspectives on geometric solvability while maintaining consistency with classical theory.