I am trying to understand the concept of embedded submanifolds and have the following understanding:
Suppose we have a smooth manifold $M$ and $N$ of dimensions $m$ and $n$ such that $m\gt n$ .Let $F\colon N\to M$ be a smooth map and a topological embedding onto its image $F(N)$ under subspace topology that $F(N)$ inherits from $M$. So, there is a smooth structure on $F(N)$ such that $F\colon N \to F(N)$ is a diffeomorphism (the smooth coordinate maps are just maps of the form $f\circ F^{-1}$, where $f$ is a smooth coordinate map for $N$). Therefore, it gives us a smooth manifold $F(N)$ of dimension similar as $N$ which sits inside $M$.
Now, the definition of embedded submanifolds as given in the text of boothby is:
image of a topological embedding+immersion is an embedded submanifold
My problem is:
If just taking $F\colon N \to F(N)$ as a topological embedding gives $F(N)$ diffeomorphic to $N$. So, it gives a smooth manifold $F(N)$ with same dimension as $N$, why do we even need to consider immersions?
Thanks in advance
Best Answer
Let $N=\Bbb R$ and $M=\Bbb R^2$. The map $F\colon N\to M$, $x\mapsto (x^3,0)$ is a topological embedding of $\Bbb R$ into $\Bbb R^2$. However, the image $F(N)=\Bbb R\times 0\subset \Bbb R^2$ does not inherit the same smooth structure as a subspace of $M$ as it does by pushing forward the smooth structure from $N$. (The two smooth structures on $F(N)$ yield diffeomorphic manifolds, but are not equivalent.) To avoid this, you want $F$ to be an immersion, so that both smooth structures (the one pushed forward via $F$ and the one inherited from $M$) are the same.
From $M$ the subspace $\Bbb R\times 0$ inherits a smooth structure with atlas defined by the global chart $(x,0)\mapsto x$. Pushed forward via $F$, we get a smooth structure on $\Bbb R\times 0$ with atlas defined by the global chart $(x,0)\mapsto x^{1/3}$. Now check that with respect to these two atlases the identity map $\Bbb R\times 0 \to \Bbb R\times 0$ is not a diffeomorphism, so the two smooth structures are different. They still define diffeomorphic manifolds, since $(x,0)\mapsto (x^3,0)$ is a diffeomorphism with respect to these two atlases.