# Asymptotic Bound on Minimum Epsilon Cover of Arbitrary Manifolds

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 $$\| x – p\|_{2}$$.

Is it the case that $$\mathcal{N}(\epsilon) \in \Theta\left(\frac{1}{\epsilon^k}\right)$$? If so, is there a reference?