round metric on the unit sphere ${\mathbb{S}}^{n}$ in normal coordinates, and the flat metric in coordinates. We will denote the round metric hereafter by ${\mathring{g}}$ and the flat metric by ${\overline{g}}$. On compact subsets of $B_{\epsilon}(0)$ all three of these metrics are comparable. Since $g_y(0,0)_{\beta \nu} = {\mathring{g}}(0)_{\beta \nu} = {\overline{g}}(0)_{\beta \nu} = \delta_{\beta \nu}$, continuity of the metrics implies we may find an $r = r(y)$ with $0 < r < \epsilon/2$ so that

-   $\frac{1}{4} {\mathring{g}}_{\beta \nu} \leq (g_Y)_{\beta \nu} \leq 4 {\mathring{g}}_{\beta \nu}$ on $B_r(0)$,

-   $\frac{1}{4} {\overline{g}}_{\beta \nu} \leq (g_Y)_{\beta \nu} \leq 4 {\overline{g}}_{\beta \nu}$ on $B_r(0)$,

-   $B_r(0) \subset B_{2r}(0) \subset B_{\epsilon}(0)$.

Compactness of $Y$ yields a finite subcover of the balls $B_{r(y)}(y)$ that cover $Y$. Label these finitely many balls $W_i$. Label the balls with the same centres and radius $2r(y)$ as $V_i$ and observe $W_i \subset \overline{W_i} \subset V_i$. We refer to the covering of $Y$ by $\{W_i\}$ as the *reference covering for Y*.

The choice of this covering ensures that distances between points in $W_i \subset Y$ with respect to the metrics $(g_Y)_i$, ${\mathring{g}}_i$, and ${\overline{g}}_i$ are all comparable. We refer to this property again as the *distance comparison principle*. \[dcp\]

In 