# Smooth map between smooth manifold and boundary of manifold

This is my first question here so I'm asking for your understanding. I also apologize in advance for my English.

I'm beginner on smooth manifolds topics and I can't solve the following problem.

Let $$f$$:$$M\rightarrow N$$ be smooth map between smooth manifolds. Let's assume that $$im (f) \subseteq \partial N$$. Prove that $$f$$ considered as map $$M \to \partial N$$ is smooth.

I would like to know how to prove it formally.
My attempt:
Let $$(U,\varphi)$$ and $$(V,\psi)$$ be smooth charts. I know that $$\psi \circ f \circ \varphi^{-1}$$ is smooth on its domain. I want to show that for chart $$(W, \theta)$$, where $$W = V \cap \partial N$$ and $$\theta = \psi|_{W}$$, the composition $$\theta \circ f \circ \varphi^{-1}$$ is smooth, but I do not know how. I will be really grateful for any help.

• $\partial N$ is an embbeded submanifold of $N$ and the image of $f$ lies on it. – DiegoMath May 28 at 0:15
• @DiegoMath Thank You for your answer, but it is not clear for me that this implies smoothness of $f$. You mean that embbeding is smooth and "this first $f$ " is a result of composition this embedding and a "second $f$", thus $f: M \rightarrow \partial N$ must be smooth? – S_J May 28 at 7:44

The proof is a follow-your-nose manipulation of the definitions of charts for $$M$$, $$N$$ and $$\partial N$$.
Let $$m = \text{dimension}(M)$$ and $$n = \text{dimension}(N)$$.
Let $$x \in M$$ and $$y=f(x) \in \partial N$$. We want to prove that $$f : M \to N$$ is smooth at $$x$$ if and only if $$f : M \to \partial N$$ is smooth at $$x$$. Since smoothness is independent of charts, it suffices to construct particular charts $$(U,\phi)$$ of $$M$$ at $$x$$, $$(V,\psi)$$ of $$N$$ at $$y$$, and $$(W,\theta)$$ of $$\partial N$$ at $$y$$, such that $$f(W) \subset U \cap V$$, and such that the map $$\psi \circ f \circ \phi^{-1}$$ is smooth if and only if the map $$\theta \circ f \circ \phi^{-1}$$ is smooth.
Start with an arbitrary chart of $$N$$ near $$y$$, given by $$\psi : V \to \mathbb H^n$$ where $$\mathbb H^n = \{(t_1,...,t_{n-1},t_n) \in \mathbb R^n \mid t_n \ge 0\}$$. Let $$W = \partial N \cap g^{-1}(V)$$. Let $$\pi : \mathbb H^n \to \mathbb R^{n-1}$$ be the projection given by $$\pi(t_1,...,t_{n-1},t_n) = (t_1,...,t_{n-1})$$. Let $$\theta : W \to \mathbb R^{n-1}$$ be given by $$\theta = \pi_n \circ g$$. As a consequence of the definition of the boundary it follows that $$\theta$$ is a chart of $$\partial N$$ near $$y$$.
Now consider a chart of $$M$$ near $$x$$, given by $$\phi : U \to \mathbb R^m$$. We may assume that $$U \subset \theta^{-1}(W)$$, by replacing $$U$$ with $$U \cap \theta^{-1}(W)$$. It also follows that $$U \subset \psi^{-1}(V)$$.
We may write the local expression for $$f$$ using the $$U$$ and $$V$$ coordinates as $$\psi \circ f \circ \phi^{-1}(x_1,...,x_m) = (f_1(x_1),...,f_{n-1}(x_1,...,x_m),f_n(x_1,...,x_m))$$ but notice that since $$f(M) \subset \partial N$$ we must have $$f_n(x_1,...,x_m)=0$$. Therefore, we may simplify the above formula to $$\psi \circ f \circ \phi^{-1}(x_1,...,x_m) = (f_1(x_1),...,f_{n-1}(x_1,...,x_m),0)$$ By composing with the projection map $$\pi$$, it follows that the local expression for $$f$$ using the $$U$$ and $$W$$ charts is given by $$\theta \circ f \circ \phi^{-1}(x_1,...,x_m) = (f_1(x_1,...,x_m),...,f_{n-1}(x_1,...,x_m))$$ It follows that $$\psi \circ f \circ \phi^{-1}$$ is smooth if and only if the functions $$f_1,...,f_{n-1}$$ are smooth if and only $$\theta \circ f \circ \phi^{-1}$$ is smooth.