Constructing Two Metrics for Spencer Cohomology: Hodge Decomposition of Constrained Bundles

Abstract: This paper establishes a metric framework for Spencer complexes based on the geometric theory of compatible pairs $(D,\lambda)$ in principal bundle constraint systems, solving fundamental technical problems in computing Spencer cohomology of constraint systems. We develop two complementary and geometrically natural metric schemes: a tensor metric based on constraint strength weighting and an induced metric arising from principal bundle curvature geometry, both maintaining deep compatibility with the strong transversality structure of compatible pairs. Through establishing the corresponding Spencer-Hodge decomposition theory, we rigorously prove that both metrics provide complete elliptic structures for Spencer complexes, thereby guaranteeing the existence, uniqueness and finite-dimensionality of Hodge decompositions. It reveals that the strong transversality condition of compatible pairs is not only a necessary property of constraint geometry, but also key to the elliptic regularity of Spencer operators, while the introduction of constraint strength functions and curvature weights provides natural weighting mechanisms for metric structures that coordinate with the intrinsic geometry of constraint systems. This theory tries to unify the differential geometric methods of constraint mechanics, cohomological analysis tools of gauge field theory, and classical techniques of Hodge theory in differential topology, establishing a mathematical foundation for understanding and computing topological invariants of complex constraint systems.