I'm looking for a proof of following theorem:
Let $M \subset \mathbb{R}^{n+k}$ a $n$-dimensional submanifold and $V \subset_{\text{ open}} \mathbb{R}^n$ open set. Let $\gamma \in C^{\infty}(V, \mathbb{R}^{n+k})$ be injective with $rank D\gamma \vert_v =n$ for every $v \in V$ and $\gamma(V) \subset M$.
Then $\gamma: V \to \gamma(V)$ is a homeomorphism.
Additionally: Is $\gamma(V)$ open in $\mathbb{R}^{n+k}$?
Let $x\in V$, consider a chart $(U, f)$ such that $\gamma(x) \in U$, you can write (since $M$ inherits the induced topology) $U =W\cap M$ where $W$ is an open subset of $\mathbb{R}^{n+k} $, let $Z=\gamma^{-1}(W)=\gamma^{-1}(U)$. $f\circ \gamma_{\mid Z} $ is an injective differential map between two open subsets of $\mathbb{R}^n$ whose rank is $n$. The local inversion theorem implies it has an inverse $g$, $ g^{-1} \circ f$ is the inverse of the restriction of $\gamma$ to $Z$.