Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds

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.