Immersion between manifolds I am currently reading various papers in the field of differential geometry and one thing has caught my attention: the authors always consider a smooth manifold $M$ and an immersion $x:M\longrightarrow\mathbb{R}^{n+1}$ and then they identify $M$ with its image $\Sigma:=x(M)$  and call this an immersed submanifold of $\mathbb{R}^{n+1}$. Going back to my lecture notes, the usual way is to parameterize $\Sigma$ for $p\in\Sigma$ by $\kappa:U\subset\mathbb{R}^{n}\longrightarrow V\subset\Sigma$, $U,V$ open. Now, choose a chart $(y,\tilde{U})$ around $x^{-1}(p)$ on $M$.
Is there a natural way to connect or identify the coordinates on $M$ induced by $y$ with those on $\Sigma$ chosen from $\kappa$? My cunfusion regarding this question arises from the fact that I wonder why we consider $\Sigma$ being the image $x(M)$, instead of simply saying that we have $\Sigma\subset\mathbb{R}^{n+1}$. Whats the benefit from this "identification" of $\Sigma$ with $M$?
 A: By Local immersion theorem, there is a co-ordinate/charts $\psi, \phi$ by which $x : M \rightarrow \mathbb{R}^{n+1}$ is an inclusion i.e., $\psi x \phi^{-1} \left(x_1,...,x_{\dim(M)} \right) = \left(x_1,...,x_{\dim(M)},0,...,0 \right)$. Here $\psi: V \subseteq x(M) \rightarrow \mathbb{R}^{\dim(M)} \times 0^{n+1-\dim(M)}$. So you can think that $\kappa = \psi^{-1}$ (after removing $0^{n+1-\dim(M)}$).
A: 
My confusion regarding this question arises from the fact that I wonder why we consider $\Sigma$ being the image $()$, instead of simply saying that we have $\Sigma\subset \mathbb{R}^{n+1}$. Whats the benefit from this "identification" of $\Sigma$ with $M$?

Well, without an immersion $x:M\xrightarrow{} \mathbb{R}^{n+1}$, the subset $\Sigma$ itself is not a manifold. As an example, consider the figure eight on the plane, which is not a manifold because at the center (double) point it isn't locally diffeomorphic to $\mathbb{R}$. But if you see it as an immersion of $S^1$, you can talk about the charts, just like the $\kappa$ you mentioned. Without the information that it is immersed, the image $\Sigma$ is necessarily a submanifold, simply because it has 'self-intersections'. Also, when you have the immersion in hand, it gives you an identification between the tangent spaces of $M$ and subspaces of $\mathbb{R}^{n+1}$. For example in the figure eight, the immersion of $S^1$ will induce two such subspaces at the center point.

Is there a natural way to connect or identify the coordinates on $M$ induced by $y$ with those on $\Sigma$ chosen from $\kappa$? ... Now, choose a chart $(,\tilde{U})$ around $x^{-1}(p)$ on $M$.

Like I said, you actually can't choose charts $\kappa$ unless you consider $\Sigma$ as an immersed submanifold. Also, $x^{-1}(p)$ is not necessarily a single point, so you can't really choose a chart around it. A useful thing would be to consider $M$ as an abstract manifold whenever immersions are involved.
