the definition of embedded submanifolds as given in the text of boothby is:
image of a topological embedding+immersion is an embedded submanifold
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 $M$.Also, F(N) has a smooth structure naturally inherited from smooth structure on M and both of these stucture might be completely different as given in the example below:
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.)
I do understand that we want to investigate meaningful ways of endowing $\pmb F(N)$ with a smooth submanifold structure. The word "submanifold" here suggests here that something about the smooth structure on $\pmb F(N)$ needs to be compatible with the smooth structure on $\pmb M$.But,as shown in example above, there are two smooth structures on $\pmb F(N)$.
My question is:
Is there a role of taking $F$ as an immersion, so that both smooth structures (the one pushed forward via $F$ and the one inherited from $M$) are the same?if not so,then why are immersions considered in definition??
Any input is welcome!
The comment by @Moishe Kohan sums up the reasons pretty well, but let me expand on them a little bit.
The most important reason for assuming $F$ is an immersion is that if you don't, then $F(N)$ can be a thing that we definitely do not want to consider as an immersed smooth submanifold. For example, consider the map $F\colon \mathbb R\to \mathbb R^2$ given by $$ F(x) = (x^3, x^2). $$ This is a topological embedding and a smooth map. Its image has a cusp at the origin:
The whole idea of a smooth submanifold is that locally, it's supposed to look like a smoothly deformed version of a linear subspace. This does not.
The second reason for insisting on an immersion is that even in the case where the image set happens to be a nice smooth submanifold, the smooth structure on $N$ is supposed to have something to do with that on $M$. Yes, you can consider the case where they don't match, such as the map $G\colon \mathbb R\to \mathbb R^2$ given by $G(x) = (x^3,0)$. The image set is the $x$-axis, but the smooth structure determined by $G$ is not related to the one it inherits from $\mathbb R^2$. In this case, if you want to think of the image as a "subobject" of $\mathbb R^2$, the original smooth structure is completely irrelevant. Requiring that the map be an immersion ensures that the smooth structure on $N$ is closely related to that of $M$.