Good afternoon!
Here is my problem. Say I have three (smooth) manifolds, $Z \subseteq Y \subseteq X$ such that $Z$ is a submanifold of $Y$ and $Y$ is a submanifold of $X$. Say $Z, Y, X$ have respective dimensions $p, m, n$.
I'm trying to prove $Z$ is a submanifold of $X$.
The definition I use for submanifold is: $M$ is a submanifold of $N$ if there exists a chart $(U,f)$ compatible with the smooth structure on $N$ such that $f(U\cap M) = f(U) \cap A$ where $A$ is an affine subspace of $\mathbb{R}^{\text{dim}(N)}$.
I've found a few similar posts ([1], [2]). [1] uses a different definition, and [2] uses the same definition as I do, but there's no answer to the post. Here is where I got with this problem:
Let $z \in Z$. We know that $Z$ is a submanifold of $Y$, so there exists a chart $(V,g)$ on $Y$, compatible with the smooth structure, such that $z \in V$ and $g(V \cap Z) = g(V)\cap A_p$, where $A_p \subseteq R^m$ is affine of dimension $p$.
Similarly, $z \in Y$ which is a submanifold of $X$, so there exists a chart $(U, f)$ on $X$ such that $z \in U$ and $f(U\cap Y) = f(U)\cap A_m$, where $A_m \subseteq \mathbb{R}^n$ is affine of dimension $m$.
Now, we'd like to find a chart $(W, h)$ on $X$ such that $x \in W$ and $h(W \cap Z) = h(W) \cap A_p$.
I was thinking of this:
Without loss of generality, we can assume $V \subseteq U$. Now, the chart $g$ goes from $V$ to $\mathbb{R^m}$. So we can compose it with a linear transformation $\tau$ so that $\tau\circ g: V \to A_m \subseteq R^n$.
Now, $(\tau \circ g)(V\cap Z) = (\tau\circ g)(V) \cap \tau(A_p)$, where $\tau(A_p)$ also is an affine subspace of dimension $p$, this time in $\mathbb{R}^n$.
The problem is, $V$ is a coordinate patch on $Y$, not a patch on $X$, so this still isn't good enough. And this is where I'm stuck.
I was thinking, since $g$ has to be compatible with the standard structure of $Y$ (which is compatible with restrictions of the standard structure on $X$), the restriction of $f$ to $V$ has to be compatible with $g$. In particular, there is a diffeomorphism $\phi := \left.f\right|_Y \circ g^{-1} : g(V) \to f(V\cap Y) \subseteq A_m \subseteq \mathbb{R}^n$.
But whatever I do, there's still the issue that we're working on a chart of $Y$, not a chart of $X$.
If someone would have an idea on how to proceed, I'd be really grateful!
Thanks, have a nice day
