Metrizability and Dynamics of Weil Bundles
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916644069048320 |
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| author | Tchuiaga, Stéphane Koivogui, Moussa Balibuno, Fidèle |
| author_facet | Tchuiaga, Stéphane Koivogui, Moussa Balibuno, Fidèle |
| contents | 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 \(π_\ast(M^\mathbf{A}) \cong π_\ast(M)\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17945 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metrizability and Dynamics of Weil Bundles Tchuiaga, Stéphane Koivogui, Moussa Balibuno, Fidèle Differential Geometry 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 \(π_\ast(M^\mathbf{A}) \cong π_\ast(M)\). |
| title | Metrizability and Dynamics of Weil Bundles |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2501.17945 |