Stroock (Essentials of Integration Theory For Analysis, $\S8.3.4$) defines a $k$-dimensional submanifold of $\mathbb{R}^n$ to be a set $M \subseteq \mathbb{R}^n$ such that for all $x \in M$, there exist $r > 0$ and a function $\phi \in C^1(B_{\mathbb{R}^k}(0;1),\mathbb{R}^n)$ with the following properties:
$\phi$ is injective.
The Jacobian of $\phi$ never vanishes.
$\phi$ takes $B_{\mathbb{R}^k}(0;1))$ onto $B_{\mathbb{R}^n}(x;2r) \cap M$
$\phi(B_{\mathbb{R}^k(0;1)}) = B_{\mathbb{R}^n}(x;2r) \cap M$.
(Here $B_{\mathbb{R}^k}(0;1)$ is the ball of radius 1 centered at 0 in $\mathbb{R}^k$.) It is clear to me that any such set $M$ is a $k$-dimensional submanifold of $\mathbb{R}^n$, but the converse is not so clear; the best I can establish is the following, based on one of Spivak's (Calculus on Manifolds, Theorem 5-2) characterizations of submanifolds. If $M$ is a $k$-dimensional submanifold of $\mathbb{R}^n$, then given $x \in M$ there are $U \subseteq \mathbb{R}^k, V \subseteq \mathbb{R}^n$ open and $\phi \in C^1(U,V)$ such that
- $\phi$ is injective.
- $\text{rank } \phi'(u) = k$ for all $u \in U$.
- $\phi(U) = V \cap M$.
- $\phi^{-1} \in C(\phi(U),U)$.
Take $r > 0$ such that $B_{\mathbb{R}^n}(p;2r) \subseteq V$ and $\delta > 0$ such that $\phi(B_{\mathbb{R}^k}(\phi^{-1}(p);\delta)) \subseteq B_{\mathbb{R}^n}(p;2r)$. Then composing $\phi$ with a translation we find a function $\psi$ such that $\psi(B_{\mathbb{R}^k}(0;1)) \subseteq B_{\mathbb{R}^n}(p;2r) \cap M$, $\psi$ is injective, and the Jacobian of $\psi$ never vanishes.
Is it possible to achieve Stroock's equality (3) from this approach or another method?