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$ ?