I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $\mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).

The definition of a submanifold I learned is:

Let $M$ be an $m$-dimensional manifold. We say that a subset $L \subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p \in L$ there exists an adapted chart $\phi: U \rightarrow V' \times V''$ with $U \subseteq L$ open in $L$, $V' \subseteq \mathbb{R}^n$ open in $\mathbb{R}^n$ and $V'' \subseteq \mathbb{R}^{m-n}$ open in $\mathbb{R}^{m-n}$ such that $\phi (U \cap L) = V' \times \{0\}$ with $0 \in \mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $\phi : U \cap L \rightarrow V'$.

The authors of the book define a submanifold of $\mathbb{R}^m$ as follows

A subset $L \subseteq \mathbb{R}^m$ is called an $n$-dimensional submanifold of $\mathbb{R}^m$ if for every point $p \in L$ there exists an open set $U \subseteq \mathbb{R}^m$ containing $p$ and an open subset $V\subseteq \mathbb{R}^m$ together with a diffeomorphism $\phi$ from $U$ to $V$ such that $\phi(M \cap U)=V \cap (\mathbb{R}^n \times \{0\})$ with $0 \in \mathbb{R}^{m-n}$.

,which directly coincides with the book's definition if one views $\mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(\mathbb{R}^m, id)$. Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $\mathbb{R}^m$ around a point $p \in L$ as follows

Let $\phi: U \rightarrow V$ be a homeomorphism from an open set $U \subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $\mathbb{R}^n$ such that $ i_M \ \circ \ \phi ^{-1}$ is a $C^\infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $\mathbb{R}^m$.

How is this equivalent to the restricted charts given in the first definition? Why does $ i_M \ \circ \ \phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $\mathbb{R}^m$. What's a good way to think about the approach taken by the book?


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