any two distinct points in $\mathbb{R}^n$ are connected by a unique line segment realizing their distance, the chord distance function $d$ is smooth away from the diagonal $\{(p,p) \in \Sigma \times \Sigma \, : \, p \in \Sigma\}$.

\[D:Cpq\] For any two distinct points $p,q$ on $\Sigma$, we will use $C_{p,q}$ to denote the unique oriented chord from $p$ to $q$. The outward unit conormal, denoted by $\eta$, is the unit vector at $\partial C_{p,q}$ tangent to $C_{p,q}$ pointing out of $C_{p,q}$. Note that $\eta(p)=-\eta(q)$. (see Figure \[F:Cpq\])

![A chord $C_{p,q}$ joining $p$ and $q$, the outward unit conormals $\eta$ and their (negative) tangential components along $\Sigma=\partial {\Omega}$[]{data-label="F:Cpq"}](Cpq.png){height="5cm"}

Let $C(t)=C_{p_t,q_t}$ be a smooth family of chords with distinct end points $p_t,q_t \in \Sigma$. If $\ell(t)=d(p_t,q_t)$ is the length of the chord $C(t)$, the first variation formula for arc length (see for example [@Cheeger-Ebin (1.5)]) implies that $$\label{E:1st-var-a} 
\frac{d \ell}{dt} =\langle \frac{dp_t}{dt}, \eta(p_t) \rangle + \langle \frac{dq_t}{dt}, \eta(q_t) \rangle. 
$$ Note that the interior integral term vanishes as $C(t)$ is a geodesic for every $t$. Since $p_t$ and $q_t$ lies on $\Sigma$ for all $t$, both $dp_t/dt$ and $dq_t/dt$ are tangential to $\Sigma$. Therefore, we can express (\[E:1st-var-a\]) as $$\label{E:1st-var-b}
\frac{d 