Context: 1. Are manifold subsets submanifolds? 2. Can manifold subsets always be made into submanifolds? 3. Why is the inclusion from a submanifold smooth?
Let $A,B$ be topological spaces with $A \subseteq B$ and $A$ a topological subspace of $B$. Suppose $A$ and $B$ become smooth manifolds $(A,\mathscr A)$ and $(B,\mathscr B)$ with respectively with dimensions $a$ and $b$.
- What's an example where the inclusion map $\iota: (A,\mathscr A) \to (B,\mathscr B)$ is smooth and a topological embedding but not an immersion?
There are a lot of examples of smooth immersions that are not topological embeddings (and thus not smooth embeddings) like this. There are also examples of smooth topological embeddings that are not immersions (and thus, again, not smooth embeddings) like this. In some of the questions linked above, there were examples that where $\iota$ wasn't smooth or even continuous. The purpose of this question is to ask specifically about the inclusion map and the case that the inclusion map is smooth. If such examples exists, then this tells me there's nothing particularly different about the inclusion map.
- What's an example where the inclusion map $\iota: (A,\mathscr A) \to (B,\mathscr B)$ is smooth but not a topological embedding?
Just checking my understanding. If there are no such examples, then (1) could simply ask "smooth but not immersion".
Best Answer
Question 1:
Let $A = \mathbb R \times \{ 0 \}$ and $B = \mathbb R^2$. Then $A$ is a a topological submanifold of $B$. We give $B$ its standard smooth structure (with single chart atlas $id : B \to \mathbb R^2$) and $A$ the smooth structure with single chart atlas $f : A \to \mathbb R, f(x,0) = \sqrt[3]{x}$. Then $\phi = id \circ f^{-1} : \mathbb R \to \mathbb R^2$ is smooth since $\phi(x) = (x^3,0)$. Thus $\iota$ is smooth. However, it is not immersion because $T_{(0,0)} \iota$ is the zero map.