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Metrizability and Dynamics of Weil Bundles

Stéphane Tchuiaga, Moussa Koivogui, Fidèle Balibuno

TL;DR

This work analyzes metrizability and dynamics on Weil bundles M^A, proving that M^A carries a canonical, complete, weighted metric $ abla d_w$ that integrates base geometry with infinitesimal directions. It develops a curve-lifting framework and shows that topological invariants and dynamics on M lift to M^A via Morimoto's vanishing lemma, with H^*(M^A) ≅ H^*(M) and π_*(M^A) ≅ π_*(M). The authors further establish a functorial lift of diffeomorphisms, endowing Diff^∞(M^A) with a natural p.w.t. topology and transferring stability properties from M to M^A. Collectively, the results demonstrate that Weil bundles enrich infinitesimal information without altering global topology, enabling transfer of dynamical properties and offering potential applications in geometric mechanics and PDE-informed latent spaces.

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 \(π_\ast(M^\mathbf{A}) \cong π_\ast(M)\).

Metrizability and Dynamics of Weil Bundles

TL;DR

This work analyzes metrizability and dynamics on Weil bundles M^A, proving that M^A carries a canonical, complete, weighted metric that integrates base geometry with infinitesimal directions. It develops a curve-lifting framework and shows that topological invariants and dynamics on M lift to M^A via Morimoto's vanishing lemma, with H^*(M^A) ≅ H^*(M) and π_*(M^A) ≅ π_*(M). The authors further establish a functorial lift of diffeomorphisms, endowing Diff^∞(M^A) with a natural p.w.t. topology and transferring stability properties from M to M^A. Collectively, the results demonstrate that Weil bundles enrich infinitesimal information without altering global topology, enabling transfer of dynamical properties and offering potential applications in geometric mechanics and PDE-informed latent spaces.

Abstract

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

Paper Structure

This paper contains 19 sections, 10 theorems, 39 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weil Algebras and Infinitesimal Structures
  4. Weil Bundles
  5. Infinitely near points
  6. Weil bundles as manifolds
  7. Local coordinates on $M^\mathbf{A}$
  8. $M^\mathbf{A}$ as an $\mathbf{A}$-manifold
  9. $M^\mathbf{A}$ as a Real-manifold
  10. Example: Coordinate transitions on $(\mathbb{R}^2)^{\mathbb{R}[\epsilon]/(\epsilon^3)}$
  11. Topologies on $M^\mathbf{A}$
  12. Weighted norms
  13. Controlling the maps $f \mapsto L_\xi(f)$
  14. Lifting of curves
  15. Dynamics
  16. ...and 4 more sections

Key Result

Proposition 2.4

Mo Let $M$ be a smooth manifold and $\mathbf A$ a Weil algebra. The collection $M^\mathbf A$ of all infinitely near points of $M$ is a smooth manifold. If $V$ is a vector space, then $V^\mathbf A\approx V\oplus_{\mathbb R} \mathcal{A}$. Furthermore, we have $(M\times N)^\mathbf A\approx M^\mathbf A\

Figures (3)

  • Figure 1: Lifting of curve from $M$ to $M^\mathbf{A}$.
  • Figure 2: Fixed points of the rotation map $\phi(\theta) = \theta + \alpha$ on $S^1$ and its Weil lifting $\phi^\mathbf{A}$ on $M^\mathbf{A}$.
  • Figure 3: The commutative diagram for $F$ : Relationship between $Diff(M)$ and $Diff(M^\mathbf{A})$.

Theorems & Definitions (24)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 14 more