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arXiv · 2509.19620

Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds

Isabel Beach·Haydeé Contreras-Peruyero·Erin Griffin·Regina Rotman·Catherine Searle
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Abstract

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.

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Isabel Beach, Haydeé Contreras-Peruyero, Erin Griffin, Regina Rotman, Catherine Searle. 2025-09-23. Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds. https://arxiv.org/abs/2509.19620

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