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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.19620 |
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| _version_ | 1866918147081109504 |
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| author | Beach, Isabel Contreras-Peruyero, Haydeé Griffin, Erin Rotman, Regina Searle, Catherine |
| author_facet | Beach, Isabel Contreras-Peruyero, Haydeé Griffin, Erin Rotman, Regina Searle, Catherine |
| contents | Let $N$ be a closed submanifold of a complete manifold, $M$. Then under certain topological conditions, there exists an orthogonal geodesic chord beginning and ending in $N$. In this paper we establish an upper bound for the length of such a geodesic chord in terms of geometric bounds on $M$. For example, if $N$ is a $2$-dimensional sphere embedded in a closed Riemannian $n$-manifold, then there exists an orthogonal geodesic chord in $M$ with endpoints on $N$ that has length at most $$ (4d+96D +8232\sqrt{A})(2n+1) $$ where $d$ is the diameter of $M$, and $A$ and $D$ are the area and intrinsic diameter of $N$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds Beach, Isabel Contreras-Peruyero, Haydeé Griffin, Erin Rotman, Regina Searle, Catherine Differential Geometry Let $N$ be a closed submanifold of a complete manifold, $M$. Then under certain topological conditions, there exists an orthogonal geodesic chord beginning and ending in $N$. In this paper we establish an upper bound for the length of such a geodesic chord in terms of geometric bounds on $M$. For example, if $N$ is a $2$-dimensional sphere embedded in a closed Riemannian $n$-manifold, then there exists an orthogonal geodesic chord in $M$ with endpoints on $N$ that has length at most $$ (4d+96D +8232\sqrt{A})(2n+1) $$ where $d$ is the diameter of $M$, and $A$ and $D$ are the area and intrinsic diameter of $N$, respectively. |
| title | Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.19620 |