The embedded submanifolds of a smooth manifold (without boundary) of codimension 0 are exactly the open submanifolds

Question

I'm reading the proof of the following proposition in Lee's book Introduction to Smooth Manifolds:

Proposition 5.1: Let $M$ be a smooth manifold. The embedded submanifolds of codimension $0$ in $M$ are exactly the open submanifolds.

Lee proves that the set of points of such manifolds $U$ (codimension $0$ in $M$) is open in $M$, but he says nothing about the smooth structure. By definition, the smooth structure of an open submanifold $V$ is determined by the smooth charts in $M$ defined on open subsets of $V$. But it seems that the smooth charts of $U$ as an embedded submanifold need not coincide with the smooth structure of $U$ as an open submanifold.

Both smooth structures must coincide, or both smooth manifolds ($U$ as an embedded submanifold and $U$ as an open submanifold) are diffeomorphic, or what is happening here?

Thank you in advance for any help.

Answer 1

There is a proposition that characterizes embedded submanifolds of codimension $k$ as follows.

Let $N\subset M$ denote a codimension $k$ embedded submanifold of $M$. If $p\in N$, then there exists a coordinate chart $(U,x^1,\ldots, x^n)$ centred at $p$ such that $$N\cap U = \{q\in U: x^{k+1}(q) = \cdots = x^n(q) = 0\}.$$

What happens when you apply this to the case of codimension $k=0$? It simply says that around a point $p$ the coordinate charts of $N$ can be constructed by restricting chart maps from those of $M$. In particular, the smooth structures are compatible.

Answer 2

Although Alekos' answer is good, I'll write an answer of my own since Chapter 5 of Lee is quite fresh in my memory, and I happen to remember that the result that Alekos quotes is Theorem 5.8, which comes after Proposition 5.1 in the book.

Given that Lee hasn't yet proven Theorem 5.8 at this point in the book, you might wonder how Lee intends his proof of Proposition 5.1 to work. The key sentence in the proof of Proposition 5.1 is this one:

The inclusion $\iota : U \to M$ is a smooth embedding by definition, and therefore it is a local diffeomorphism by Proposition 4.8.

And here is the relevant part of Proposition 4.8:

Suppose $M$ and $N$ are smooth manifolds and $F: M \to N$ is a map. If $\text{dim}(M) = \text{dim}(N)$ and $F$ is a smooth immersion, then $F$ is a local diffeomorphism.

In fact, we can make a bolder statement. $\iota : U \to M$ is injective, and it is a local diffeomorphism. So $\iota : U \to \iota(U)$ must be a diffeomorphism! (I'm abusing notation here, using $\iota$ to represent two different maps with different codomains. Please forgive me. Of course, in the second map, the codomain $\iota(U)$ is viewed as an open submanifold of $M$.)

Now let me summarise very precisely what we've learnt up to this point, following Lee's definition of embedded manifolds as closely as possible.

Suppose $U$ is a codimension-zero embedded submanifold of $M$. That is, suppose that $U$ is a subset of $M$, endowed with the subspace topology induced by the inclusion $\iota : U \to M$, and endowed with a smooth structure $\mathcal A_{\text{embedded}}$ which (i) is compatible with the subspace topology, (ii) makes $U$ into a manifold whose dimension is equal to the dimension of $M$, (iii) makes the inclusion map $\iota : U \to M$ into a smooth embedding.

Then $\iota(U)$ is an open subset of $M$, and $\iota: U \to \iota(U)$ is a diffeomorphism.

Now let's think clearly about what exactly we mean when we say that $\iota: U \to \iota(U)$ is a diffeomorphism.

• The domain $U$ is viewed as a manifold with smooth structure $\mathcal A_{\text{embedded}}$.
• The codomain $\iota(U)$ is viewed as an open submanifold of $M$. Its smooth structure is $\mathcal A_{\text{open}}$: the smooth structure that contains precisely those charts from the smooth structure on $M$ whose domains lie in $\iota(U)$.
• $\iota : U \to \iota(U)$ is a diffeomorphism with respect to these two smooth structures.

Being a diffeomorphism, $\iota$ induces a bijective correspondence between the charts in $\mathcal A_{\text{embedded}}$ and the charts in $\mathcal A_{\text{open}}$: for every chart $(W, \phi)$ in $\mathcal A_{\text{embedded}}$, there is a corresponding chart in $\mathcal A_{\text{open}}$ of the form $(\iota(W), \phi \circ \iota^{-1})$; furthermore, all charts in $\mathcal A_{\text{open}}$ can be obtained from charts in $\mathcal A_{\text{embedded}}$ in this way.

But $\iota$ is really the identity map, and $\iota(U)$ is just $U$. If we now view both $A_{\text{embedded}}$ and $A_{\text{open}}$ as smooth structures on $U$, then what we've proved is that the charts in $\mathcal A_{\text{embedded}}$ are precisely the charts in $\mathcal A_{\text{open}}$. In other words, $A_{\text{embedded}}$ and $\mathcal A_{\text{open}}$ are the same.

I know this has been a slog and Alekos' answer is excellent. Still, I think there is value in wrapping one's head around the original argument in the book.