\leq 1 \right\}$$ where $a_1,\cdots,a_n$ are distinct positive real numbers.

In [@Bos63], Bos generalized Lusternik-Schnirelmann’s result to the setting of Riemannian (or even Finsler) manifolds.

\[T:Bos\] A compact Riemannian manifold $(M^n,g)$ which is homeomorphic to the closed unit ball in ${\mathbb{R}}^n$ with locally convex boundary contains at least $n$ orthogonal geodesic chords.

Moreover, he showed that the convexity assumption cannot be dropped even in ${\mathbb{R}}^2$ (see Figure \[F:Bos\]).

![Bos’ example of a non-convex domain $\Omega$ in ${\mathbb{R}}^2$ which does not have any orthogonal geodesic chord contained in $\Omega$.[]{data-label="F:Bos"}](Bos.png){height="4cm"}

Nonetheless, one can still ask for the existence of orthogonal geodesic chords, by allowing them to go *outside* the domain. This problem was first introduced by Riede [@Riede68], where he studied the variational calculus of the space $\Gamma$ consisting of piecewise smooth curves in a complete Riemannian manifold $(M^n,g)$ with end points lying on a compact submanifold $\Sigma^k \subset M$. In particular, he estimated the minimum number of critical points, which are orthogonal geodesic chords, in terms of certain topological invariant called the “cup-length” of the equivariant cohomology of $\Gamma$ with respect to the ${\mathbb{Z}}_2$-action reversing the orientation of a curve. In [@Hayashi82], Hayashi computed the cup-length when $\Sigma$ is a compact 