Metrizability and Dynamics of Weil Bundles

Abstract: This paper bridges synthetic and classical differential geometry by investigating the metrizability and dynamics of Weil bundles. For a smooth, compact manifold (M) and a Weil algebra (\mathbf{A}), we prove that the manifold (M^\mathbf{A}) of (\mathbf{A})-points admits a canonical, complete, weighted metric (\mathfrak{d}w) that encodes both base-manifold geometry and infinitesimal deformations. Key results include: (1) Metrization: (\mathfrak{d}_w) induces a complete metric topology on (M^\mathbf{A}). (2) Path Lifting: Curves lift from (M) to (M^\mathbf{A}) while preserving topological invariants. (3) Dynamics: Fixed-point theorems for diffeomorphisms on (M^\mathbf{A}) connected to stability analysis. (4) Topological Equivalence: (H^*(M^\mathbf{A}) \cong H^*(M)) and (\pi\ast(M^\mathbf{A}) \cong \pi_\ast(M)).