Metrizability and dynamics of Weil bundles

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 \(\textbf{A}\), we prove that the manifold \(M^\textbf{A}\) of \(\textbf{A}\)-points admits a canonical, weighted metric \(\mathfrak {d}_w\) that encodes both base-manifold geometry and infinitesimal deformations. Our approach relies on constructions and methods of local and global analysis. Key results include: (1). Metrization: \(\mathfrak {d}_w\) induces a complete metric topology on \(M^\textbf{A}\). (2). Path Lifting: Curves lift from \(M\) to \(M^\textbf{A}\) while preserving topological invariants. (3). Dynamics: Fixed-point theorems for diffeomorphisms on \(M^\textbf{A}\) connected to stability analysis. (4). Topological Equivalence: \(H^*(M^\textbf{A}) \cong H^*(M)\) and \(\pi _*(M^\textbf{A}) \cong \pi _*(M)\).