Let $F:N\to M$ be a smooth map from a smooth manifold $N$ to a smooth manifold $M$ (without boundary) and $S$ an immersed submanifold of $M$ such that $F(N)\subset S$. If $F$ is continuous as a map from $N$ to $S$, then $F:N\to S$ is smooth.
This is theorem $5.29$ in John Lee's introduction to smooth manifolds $2$nd edition and in that same book he wrote the following proof (I will not give the full proof but only upto the part where I don't understand):
"$Proof$: Let $p\in N$ and $q=F(p)\in S$. Then as $S$ is an immersed submanifold, it is locally embedded. Therefore there is a neighborhood $V$ of $q$ such that $V$ is embedded in $M$, that is the inclusion $i|_V:V\hookrightarrow M$ is a smooth embedding. This implies $V$ satisfies the local slice criterion, that is there exist a smooth chart $(W, \psi)$ for $M$ that is a slice chart for $V$ centerd at $q$. The fact that $(W,\psi)$ is a slice chart means that $(V_0, \widetilde{\psi})$ is a smooth chart of $V$, where $V_0 = W\cap V$ and $\widetilde{\psi}= \pi\circ \psi|_{V_0}$, with $\pi: \mathbb{R}^n\to \mathbb{R}^k$ the projection onto the first $k=\dim S$ coordinates."
I understand everything upto this line. The next line below is where my confusion started.
"Since $V_0 = \left( i|_V\right)^{-1}(W)$ is open in $V$, it is open in $S$ in its given topology, and so $(V_0, \widetilde{\psi})$ is also a smooth chart for $S$."
I know that $V_0$ is open in $S$ and I know that $(V_0, \widetilde{\psi})$ will be a (continuous) chart but I dont understand the claim that $(V_0,\widetilde{\psi})$ is a smooth chart for S. How do we make sure that $(V_0, \widetilde{\psi})$ belongs to the smooth structure of $S$ by merely knowing $V_0$ is open? Is there a relationship between the smooth structure of $V$ and the smooth structure of $S$. Please help me I'm so confuse.
Best Answer
Personally I do not like the concept of an immersed submanifold as presented by Lee and other authors. The situation is this:
We have an injective immersion $j : P \to M$. This induces a unique topology and a unique smooth structure on the image set $S = j(P)$ such that $\bar j : P \stackrel{j}{\to} S^*$ becomes a diffeomorphism, where $S^*$ denotes the set $S$ endowed with the appropriate topology and smooth structure. This $S^*$ is called an immersed submanifold of $M$.
The problem here is that $j$ is in general no homeomorphism from $P$ onto its image $j(P)$ with the subspace topology inherited from $M$. Thus, although we call $S^*$ an immersed submanifold, it is in general not even a topological subspace of $M$. In my opinion this is somewhat confusing, but of course it is a matter of taste.
However, even if $S^*$ should not be a topological subspace of $M$ globally, it is always locally one. This means that each $q \in S^*$ has an open neigborhood $V$ in $S^*$ which is an emdedded submanifold of $M$. Note that $V$ is open in $S^*$, but in general not open in $S$ with the subspace topology inherited from $M$.
In my opinion a better formulation of Lee's Theorem 5.29 would be this:
Lee's theorem is a corollary. We are given an immersed submanifold $S^*$ of $M$ and a smooth $F : N \to M$ such that $F(N) \subset S^*$ and $\bar F : N \stackrel{F}{\to} S^*$ is continuous. This $S^*$ is associated to an injective immersion $j : P \to M$ such that $\bar j : P \stackrel{j}{\to} S^*$ is a diffeomorphism. The function $f = (\bar j)^{-1} \circ \bar F : N \to P$ is continuous and $j \circ f = F$ is smooth, thus $f$ is smooth. Hence $\bar F = \bar j \circ f$ is smooth.
Our above theorem can be proved exactly as Lee's Theorem 5.29. Let us now see why $(V_0, \widetilde{\psi})$ is a smooth chart for $S^*$.
Since $V$ is open in $S^*$ and $V$ is an embedded submanifold of $M$, it suffices to show that $(V_0, \widetilde{\psi})$ is a smooth chart for $V$. But this follows from a theorem in Lee's book which says that each subspace $E \subset M$ which is an embedded submanifold has a unique smooth structure such that the inclusion map $E \to M$ is a smooth embedding. A smooth atlas on $E$ generating this smooth structure is given precisely by the collection of all $(W_0 = W \cap E, \pi \circ \psi \mid_{W_0})$ where the $(W,\psi)$ are smooth charts for $M$ which are slice charts for $E$ centered at the points $q \in E$.
Perhaps it is somewhat confusing that the smooth structure on $V$ is primarily induced by $j_V : j^{-1}(V) \stackrel{j}{\to} V$. However, since we know that $V$ is an embedded submanifold of $M$, this smooth structure agrees with the above smooth structure induced by slice charts.