Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$ of dimenion $n$.
Is it true that there exists a constant $C$ such that $$d(x,y) \leq C|x-y|$$ for all $x, y \in M$? Here $d$ is the Riemannian distance function and $|\cdot|$ is the usual norm on $\mathbb{R}^{n+1}$.
I think it is trivially true that $|x-y| \leq d(x,y)$ since the straight line minimises the distance between two points. But the reverse, I don't know.
Define the function $$ f(x,y) := \begin{cases} \frac{d(x,y)}{|x-y|} & \text{ if } x \ne y \\ 1 & \text{ if } x=y \end{cases} $$ for $(x,y) \in M \times M$. Then $f$ is continuous on a compact set, so it has a maximum which works as $C$ in your question.