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 $\| x - p\|_{2}<\epsilon$.

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

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; that is for every point $x \in M$ there exists a $p \in P$ such that $\| x - p\|_{2}<\varepsilon$.

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

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?

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 $\| x - p\|_{2}<\epsilon$.

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