Metrizability and Dynamics of Weil Bundles

TL;DR

This paper explores the geometric and dynamic properties of Weil bundles over manifolds, establishing a canonical metric structure and analyzing their topological and dynamical features.

Contribution

It introduces a canonical, complete, weighted metric on Weil bundles and investigates their topological and dynamical properties, bridging synthetic and classical differential geometry.

Findings

Weil bundles admit a canonical, complete, weighted metric.

Curves can be lifted from the base manifold to Weil bundles while preserving invariants.

Fixed-point theorems relate to stability analysis on 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 \(\pi_\ast(M^\mathbf{A}) \cong \pi_\ast(M)\).