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: $H^{\ast }(M^{\mathbf{A}})\cong H^{\ast }(M)$ and ${\pi }_{\ast }(M^{\mathbf{A}})\cong {\pi }_{\ast }(M)$.