Confusion regarding the construction of a parameterization for a manifold

Question

I am trying to prove that the differential of the inclusion map $i: X \rightarrow Y$ of a submanifold $X \subset Y$ $$di_x: T_x(X) \rightarrow T_x(Y)$$ is the inclusion map. This question has been asked here before; however, the choice of parametrization in the answer is a bit confusing.

Here is the claim reworded based on my understanding: Let $Y \subset \mathbb{R}^N$ be a $l$-dimensional manifold and $X \subset Y$ a $k$-dimensional manifold. For some neighborhood $x \in U$ in $Y$ we may choose a coordinate system diffeomorphism $\phi: U \rightarrow \mathbb{R}^l$ such that the image of $U \cap X$ is diffeomorphic to the canonical inclusion of $\mathbb{R}^k \subset \mathbb{R}^l$.

Why is this true? Here is my take. What we know is that there is a diffeomorphism $\phi: U \rightarrow \mathbb{R}^l$ such that the image of $U \cap X$ under this map is diffeomorphic to $\mathbb{R}^k$ - this is because for some sufficiently small open set $V \subset U$, $V \cap X \simeq \mathbb{R}^k$ (by $X$ being a $k$-manifold) and we can take $U$ such that $U \simeq V$. Hence, $U \cap X \simeq V \cap X \simeq \mathbb{R}^k$. Now we apply a linear transformation on $\phi$ such that the points within $V \cap X$ that represent a basis for $\mathbb{R}^k$ map to the canonical basis of $\mathbb{R}^k$ as a vector subspace of $\mathbb{R}^l$. Is this correct?

Answer 1

The solution you've cited assumes the local immersion theorem, which comes a few pages later (p. 15) in Guillemin & Pollack. Not knowing the specific context, it's the first thing that comes to mind for most of us, although in some sense it begs the question (how do we know the inclusion is an immersion?).

My suggestion is to use the only tool you have to identify the inclusion map: It is the restriction of the identity map $Y\to Y$. So just check that, more generally, if $F\colon Y\to Z$ is a smooth map and $X$ is a submanifold of $Y$, then for any $p\in X$, the derivative of the restriction of $F$ at $p$ is the restriction of the derivative. As a major hint, using parametrizations, reduce to the case that $Y=\Bbb R^\ell$ and $Z=\Bbb R^m$, with $X\subset\Bbb R^\ell$. Then if $\varphi$ is a parametrization of $X$ and $f$ is the restriction of $F$ to $X$, we have $F\circ\varphi = f\circ\varphi$, and now you can apply the chain rule.

COMMENT: This argument shows (without invoking the local immersion theorem as I have usually done in class teaching this book numerous times) that the original definition of smoothness (local smooth extensions, p. 2) gives the same definition of the derivative in this situation as using parametrizations (p. 10).