Topological Charge Conservation and Degree Rigidity for $S^{3}$-Valued Fields on Product Spacetimes

This paper develops a purely geometric formulation of topological charge conservation for smooth ( S^{3} )-valued fields defined on product spacetimes of the form ( \mathbb{R} \times \Sigma ), where ( \Sigma ) is a closed, oriented three-manifold. Using only de Rham cohomology, orientation, and the existence of a normalized volume form on ( S^{3} ), the work shows that the degree (winding number) of the field restricted to spatial slices is invariant along the product direction. This conservation law is established without reference to a metric, equations of motion, or variational principles, isolating its fundamentally topological origin. Interpreting ( S^{3} ) as ( SU(2) ), the paper relates the degree of the field to the Chern–Simons invariant of the associated flat connection ( A = n^{-1} dn ), providing a clear gauge-theoretic interpretation of topological charge. A rigidity result is also proved, demonstrating that any change in degree necessarily requires either singularities in the field or a failure of the global product structure of the domain. The results clarify the structural independence of topological charge conservation from dynamical assumptions and place it firmly within the framework of differential topology and characteristic classes.