In my differential geometry lectures, we defined an embedded submanifold in the following way:
Let $N \subset M$ and $i: N \rightarrow M$ be the inclusion map. Then we say that $N$ is an embedded submanifold of $M$ if the following hold:
i) $i$ is smooth
ii) The differential $di|p$ is injective at all $p\in N$
iii) $i$ is a homeomorphism onto its image
And iii) is dropped for an immersed manifold.
The problem is, I am finding it hard to see how the inclusion map from a subset can fail to have injective differential. In that case, why explicitly specify point ii)?
I am wondering whether it has to do with the later definition of embedded manifolds- when $N$ is not a subset of $M$. We let $\phi: N \subset M$ be an injective map and said that $\phi$ embeds $N$ in $M$ if i) - iii) hold. I understand that injective maps can fail to have injective differential, as given in these examples for instance.
So I would appreciate if someone could clarify if the inclusion map trivially has injective differential.
There is a useful way to think about the differential of a smooth function $f : M \to N$: namely in terms of curves. If you have a tangent vector $v$ at $p \in M$, then there is some curve $\gamma : I \to M$ whose tangent vector is $v$. You can find $w = df|_p (v)$ by mapping this curve via $f$ to $N$, and evaluating $w$ as the tangent vector to $f \circ \gamma$ at $f(p).$
Now, if $f$ is an inclusion map, all these curves are mapped to "themselves", and it becomes obvious that the differential is injective at all points.