$c'(1)$ are normal to $\Sigma$ at $c(0)$ and $c(1)$ respectively.

An *orthogonal geodesic chord* is also called a *free boundary geodesic* [@Zhou16] or a *double normal* [@Kuiper64] in the literature. Note that in case $\Sigma \subset {\mathbb{R}}^n$ is an embedded hypersurface which bounds a domain $\Omega$ in ${\mathbb{R}}^n$, our definition of orthogonal geodesic chords does not require the chord to be contained inside $\overline{\Omega}$ as for example in [@Giambo-Giannoni-Piccione14]. The problem of the existence of multiple orthogonal geodesic chord for submanifolds in ${\mathbb{R}}^n$ was first treated by Riede [@Riede68] as follows. Let $\mathcal{C}_\Sigma$ be the space of all piecewise smooth curve $c:[0,1] \to {\mathbb{R}}^n$ with end points on $\Sigma$, endowed with the compact open topology. There exists a ${\mathbb{Z}}_2$-action on $\mathcal{C}_\Sigma$ by $c(t) \mapsto c(1-t)$ whose fixed point set is denoted by $\Delta'$. Denote by $H^{{\mathbb{Z}}_2}_*(\mathcal{C}_\Sigma,\Delta')$ and $H^*_{{\mathbb{Z}}_2}(\mathcal{C}_\Sigma)$ the ${\mathbb{Z}}_2$-equivariant homology groups (relative to $\Delta'$) and cohomology groups respectively. The following result is taken from [@Riede68 Satz (5.5)].

\[L:Riede\] If there exists $\beta \in H^{{\mathbb{Z}}_2}_*(\mathcal{C}_\Sigma,\Delta')$ and $\alpha_1,\cdots,\alpha_s \in H^*_{{\mathbb{Z}}_2}(\mathcal{C}_\Sigma)$ (not necessarily distinct) with deg $\alpha_i>0$ for all $i$ such that $(\alpha_1 \cup \cdots \cup \alpha_s) \cap \beta \neq 0$, then there exists at least $s+1$ orthogonal geodesic chords for $\Sigma$.

