A submanifold of a submanifold is a submanifold is the ambient space.

Definition: Let $$X$$ be a manifold and $$Y \subset X$$. We say that $$Y$$ is a submanifold of $$X$$ if for every $$y \in Y$$, there exists some chart $$(U, \phi)$$ and $$p \leq n$$ such that $$\phi(U \cap Y) = \phi(U) \cap (\mathbb{R}^p \times \{0\}^{n - p})$$ where $$n$$ is the dimension of $$X$$.

Given this definition, I'm trying to prove that if $$Z' \subset Z \subset X$$ such that $$Z'$$ is a submanifold of $$Z$$ and $$Z$$ is a submanifold of $$X$$, then $$Z'$$ is a submanifold of $$X$$.

So far I've been quite unsuccessful, though I've come up with reasonable arguments but far from being rigorous. Any help greatly appreciated.

• It would be useful if you indicated your reasonable but unrigorous arguments in your question. – Travis Willse Oct 4 '18 at 12:47

Let $$Y$$ be a submanifold of $$X$$ and let $$Z$$ be a submanifold of $$X$$. Let $$n , m , l \in \mathbb{N}$$ and we suppose that $$X$$ is a $$n$$-dimensional differentiable manifold, $$Y$$ is a $$m$$-dimensional differentiable manifold and $$Z$$ is a $$l$$-dimensional differentiable manifold. Let $$p \in Z \subset Y \subset X$$. The rangs of the inclusions $$j : Y \to X \qquad \mbox{ and } \qquad k : Z \to Y$$ in $$p$$ have to be $$m$$ and $$l$$, respectively, and we will see that the rang of $$i = j \circ k : Z \to X$$ is less or equal to $$l$$ and we will be done, as $$l \leq m \leq n$$.
On the one hand, we know that there exist a chart $$(U , \varphi)$$ in $$X$$, with $$p \in U$$, and a chart $$(V_1 , {\psi}_1)$$ in $$Y$$, with $$p \in V_1$$, such that $$rg(d{(\varphi \circ j \circ {\psi}_1^{- 1})}_{{\psi}_1(p)}) = n\mbox{.}$$ On the other hand, we know that there exist a chart $$(V_2 , {\psi}_2)$$ in $$Y$$, with $$p \in V_2$$, and a chart $$(W , \phi)$$ in $$Z$$, with $$p \in W$$ such that $$rg(d{({\psi}_2 \circ k \circ {\phi}^{- 1})}_{\phi(p)}) = l\mbox{.}$$ Let $$V = V_1 \cap V_2$$ and $$\psi = {\psi}_1\big|_V$$. Using the chain rule, $$d{(\varphi \circ i \circ {\phi}^{- 1})}_{\phi(p)} = d{(\varphi \circ j \circ {\psi}^{- 1} \circ \psi \circ k \circ {\phi}^{- 1})}_{\phi(p)} =$$ $$= d{(\varphi \circ j \circ {\psi}^{- 1})}_{\psi(p)} d{(\psi \circ k \circ {\phi}^{- 1})}_{\phi(p)}\mbox{.}$$ Then $$rg(d{(\varphi \circ i \circ {\phi}^{- 1})}_{\phi(p)}) = rg(d{(\varphi \circ j \circ {\psi}^{- 1})}_{\psi(p)} d{(\psi \circ k \circ {\phi}^{- 1})}_{\phi(p)}) \leq l$$ because the rang of the product of two matrix is less or equal to the rang of both rangs.