\ell}{dt} = \langle \frac{dp_t}{dt}, \eta^T(p_t) \rangle + \langle \frac{dq_t}{dt}, \eta^T(q_t) \rangle$$ where $(\cdot)^T$ denotes the tangential component of a vector relative to $\Sigma$. More precisely, if $\pi_x:{\mathbb{R}}^n \to T_x\Sigma$ is the orthogonal projection onto the tangent space $T_x \Sigma \subset {\mathbb{R}}^n$, then $v^T=\pi_x(v)$ for any vector $v \in T_x {\mathbb{R}}^n \cong {\mathbb{R}}^n$.

It is natural to consider the (negative) gradient flow to the chord length functional, which leads to the following definition.

\[D:CSF\] A smooth family of curves $$C(u,t):I \times [0,T) \to \mathbb{R}^n$$ is a solution to the *chord shortening flow* (relative to $\Sigma$) if for all $t \in [0,T)$, we have

-   $p_t:=C(0,t)$ and $q_t:=C(1,t)$ lies on $\Sigma$,

-   $C(t):=C(\cdot,t):I \to \mathbb{R}^n$ is a constant speed parametrization of $C_{p_t,q_t}$,

-   $$\frac{\partial C}{\partial t} (0,t) = - \eta^T(C(0,t)) \qquad \text{and} \qquad \frac{\partial C}{\partial t} (1,t) = - \eta^T(C(1,t)).$$

Let us begin with some basic examples of the chord shortening flow as defined in Definition \[D:CSF\].

\[ex:flat\] Let $\Sigma$ be an affine $k$-dimensional subspace in ${\mathbb{R}}^n$. The chord shortening flow with respect to $\Sigma$ will contract any initial chord $C(0)=C_{p,q}$ to a point in finite time. The end points would move towards each other with unit speed along the chord $C(0)$ 