end points of $C(t)$. Here, $(\cdot)^N$ denotes the normal component of a vector relative to $\Sigma$ and $A:T\Sigma \times T\Sigma \to N\Sigma$ is the second fundamental form of $\Sigma$ defined by $A(u,v):=(D_u v)^N$.

Let $C(u,t):I \times [0,T) \to \mathbb{R}^n$ be a solution to the chord shortening flow relative to $\Sigma$. Since $C(t)=C( \cdot, t):I \to \mathbb{R}^n$ is a family of chords which are parametrized proportional to arc length, $\frac{\partial}{\partial t}$ is a Jacobi field (not necessarily normal) along each chord which can be explicitly expressed as $$\frac{\partial }{\partial t} = -(1-u) \,  \eta^T(0) - u \, \eta^T(1),$$ where $\eta$ is the outward unit conormal for $C(t)$. Since $[\frac{\partial}{\partial u}, \frac{\partial }{\partial t}]=0$, we have $$\label{E:commutator}
D_{\frac{\partial}{\partial t}} \frac{\partial }{\partial u}=D_{\frac{\partial}{\partial u}} \frac{\partial }{\partial t} = \eta^T(0)-\eta^T(1).$$ Moreover, as $C(t)$ is parametrized with constant speed, we have $\| \frac{\partial}{\partial u}\|= \ell$, thus $$-\eta(0)= \frac{1}{\ell} \left.\frac{\partial }{\partial u} \right|_{u=0} \quad  \text{and} \quad \eta(1)=\frac{1}{\ell} \left. \frac{\partial }{\partial u} \right|_{u=1}.$$ Fix $u=0$. Let $p=C(0,t) \in \Sigma$ and $\{e_1,\cdots, e_k\}$ be an orthonormal basis of $T_p \Sigma$ such that $(D_{e_i} e_j(p))^T=0$ for $i,j=1,\cdots,k$. Therefore, we have $$\label{E:A}
D_{\frac{\partial}{\partial t}} e_i = -A(\eta^T,e_i).$$ Using Lemma \[L:length-evolution\], (\[E:commutator\]) and (\[E:A\]), we have: $$\begin{aligned}
\frac{\partial \eta^T}{\partial t} &=& 