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 $\mathbf{A}$, 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 $\pi_\ast(M^\mathbf{A}) \cong \pi_\ast(M)$.