Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is for every point $x \in M$ there exists a $p \in P$ such that $\| x - p\|_{2}$. My goal is to show that $\mathcal{N}(\epsilon) \in \Theta\left(\frac{1}{\epsilon^k}\right)$. Note that the bound depends on the dimension $k$, not the embedding space $d$.
To this end, I'd like to construct an $\epsilon$-cover in each coordinate chart $(U, \phi)$ of $M$, where $\phi: U \subset \mathbb{R}^k \rightarrow M$. A cover in $U$ can be transformed under $\phi$ into a cover in $M$. If $\phi$ is $L$-Lipschitz, that is $\|\phi(x) - \phi(y)\|_2 \leq L \|x - y\|_{2}$, then I can create an $\epsilon$-cover over a subset $\phi(U) \subset M$, with slightly larger balls than those in $U \subset \mathbb{R}^k$ as determined by the Lipschitz constant $L$.
To have each coordinate chart $(U, \phi)$ be $L$-Lipschitz I'm willing to assume $M$ has bounded curvature. Can I subdivided $M$ into a set of coordinate charts which are $L$-Lipschitz and how many such charts do you need as a function of the curvature of $M$?
