Stationaere Kurven auf endlichdimensionalen Mannigfaltigkeiten
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908754965954560 |
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| author | Starke, Tobias |
| author_facet | Starke, Tobias |
| contents | In this work we discuss the notion of stationary curves of the length functional, the so-called (weak) geodesics, on a Riemannian manifold. The motivation behind this work is to give a detailed description of many key concepts from differential geometry that one needs in order to understand the important notion of a (weak) geodesic. For this, we mainly focus on finite-dimensional smooth manifolds, so that we can develop an intuitive and geometric understanding of the concepts that we want to discuss. At the end of this work, we also provide a rough description of how one can generalise these ideas into infinite dimensions and how one can use (weak) geodesics in special algorithms for image matching (see [21]). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05695 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stationaere Kurven auf endlichdimensionalen Mannigfaltigkeiten Starke, Tobias Differential Geometry 53A04 In this work we discuss the notion of stationary curves of the length functional, the so-called (weak) geodesics, on a Riemannian manifold. The motivation behind this work is to give a detailed description of many key concepts from differential geometry that one needs in order to understand the important notion of a (weak) geodesic. For this, we mainly focus on finite-dimensional smooth manifolds, so that we can develop an intuitive and geometric understanding of the concepts that we want to discuss. At the end of this work, we also provide a rough description of how one can generalise these ideas into infinite dimensions and how one can use (weak) geodesics in special algorithms for image matching (see [21]). |
| title | Stationaere Kurven auf endlichdimensionalen Mannigfaltigkeiten |
| topic | Differential Geometry 53A04 |
| url | https://arxiv.org/abs/2601.05695 |