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?
It is clear that restrictions from the previous definition satisfy the second definition of charts. Conversely, suppose that $\phi: U \to V$ is as in the second definition: we need to extend $\phi$ as a diffeomorphism on a neighborhood of $p$ in $\mathbb R^m$. For simplicity, let $\psi:=i_M \circ \phi^{-1}$. By the immersion theorem, up to restricting $U$ and $V$, there are diffeomorphisms $f: \tilde U \to \mathbb R^m$ and $g: V \to \mathbb R^n$ such that $\psi(x)=f^{-1}(g(x),0)$, where $\tilde U$ is an open subset of $\mathbb R^m$ such that $\tilde U \cap L=U$. Now it is enough to extend $\psi$ by $\psi(x,t):=f^{-1}(g(x),t)$, for $t \in \mathbb R^{m-n}$ in a ball of small enough radius. You can check that $\psi$ is a diffeomorphism onto its image, and that $\psi^{-1}$ is the required extension of $\phi$.
Answer: because it makes the proof above work. If it is not, here's a counter example. Take $m=2$, $n=1$, and $L=\{ (x,y) : x^2=y^3 \}$. Take $U=L$ and $\phi: L \to \mathbb R$ be defined by $\phi(x,y)=x^{1/3}$. Then $i_M \circ \phi^{-1}(t)=(t^3, t^2)$ is $C^\infty$ but not an immersion at $t=0$. If you look at a picture of $L$, you will notice that there is a cusp at $(x,y)=(0,0)$: it is not a smooth submanifold of $\mathbb R^2$ (although it is a topological submanifold, homeomorphic to $\mathbb R$).