Ytukyg

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 $Vsubseteq mathbb{R}^m$ 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?