Intuitive meaning of immersions I have a hard time understanding the concept of immersions. In my course, it was only introduced by the immersion theorem wich says:
Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be $C^{1}$ on an open subset $U \subset \mathbb{R}^{m}$. Let $a \in U$ rank$(Df_{(a)}) = m$ then there exists a neigbourhood $V$ of $f(a)$ and a diffeomorphism: $\psi: V \rightarrow \psi(V)$ such that:
$$\psi ( f(x_{1},...,x_{n})) = (x_{1},...,x_{n},\underbrace{0,...0}_\text{n-m})$$ For $x$ close enough to $a$.
The way I see it is that a 'surface' in $\mathbb{R}^{m}$ can be gradually deformed to a flat surface in $\mathbb{R}^{n}$.
Is this a reasonable way of thinking of immersions?
If so, consider:
$$f:\bar{B}_{1}(0) \subset \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}:(x,y) \rightarrow (x,y,\sqrt{1-x^{2}-y^{2}})$$
wich maps $\{ (x,y) \in \mathbb{R}^{2}| x^{2}+-y^{2} \leq 1 \}$ to the upper halve of a sphere in $\mathbb{R}^{3}$.
The differential of $f$ in $(0,0)$ is:
$$Df_{0} = \begin{pmatrix}
   1 & 0 \\
  0 & 1  \\
  0 & 0
 \end{pmatrix}$$
So the immersiontheorem says that in a neighbourhood of the 'north pole' of the upper sphere the surface can be flattened by a function $\psi$.
Consider now the function:
$$g:\mathbb{R}^{2} \rightarrow \mathbb{R}^{3}: (\theta,\phi) \rightarrow (\sin\theta  \cos \phi,\sin\theta  \sin \phi, \cos \theta)$$
Wich also represents a sphere. (not injective).
The differential in $(0,\phi)$ is now:
$$Dg_{0} = \begin{pmatrix}
   \cos \phi &  0\\
  \sin \phi & 0  \\
  0 & 0
 \end{pmatrix}$$
In this case rank$\left (Dg_{(0,\phi)} \right)<2$ so the immersion theorem doesn't apply.
My feeling is that $g$ can't be flattened near $(0,\phi)$ because the immersion theorem doesn't say it can be done. Is this true (I'm aware of the immersion theorem not being if and only if)? and why is it? Because the surface of $g$ and $f$ are the same near those points.
Also: What does immersions make important?
Bear in mind that I haven't learned anything of manifolds.
I looked at another question , but I don't have enough knowledge to understand it.
 A: The main point with immersions is how they relate to embeddings. The latter are diffeomorphic copies: $h:M\hookrightarrow\mathbb R^n$ is an embedding if it gives a diffeomorphism from $M$ onto $N=h(M)\subset\mathbb R^n$ and $N$ is a submanifold of $\mathbb R^n$ (one could use any other manifold instead of $\mathbb R^n$). Thus there is no diff-top distinction among the initial $M$ and the copy $N=h(M)$. Now for immersions this is true only locally at the source:
If $f:M\to\mathbb R^n$ is an immersion at $a\in M$ say, there is an open nbhd $U$ of $a$ in $M$ such that $f|_U:U\to\mathbb R^n$ is an embedding, that is, $U$ is diffeomorphic to $f(U)$, which is a submanifold of $\mathbb R^n$. And here comes the important thing to notice: $f(U)$ need not be open in $f(M)$, hence we cannot deduce $f(M)$ is a manifold near $f(a)$. This obstruction occurs even if $f$ is injective. This is the famous example:

(which I borrow from Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}^2$ is closed). Here
$$
f:\mathbb R\to\mathbb R^2:t\mapsto\big(\frac{t^3}{1+t^4},\frac{t}{1+t^4}\big).
$$
The image $f(\mathbb R)$ is a lemniscate with two branches crossing at the origin $(0,0)=f(0,0)$, at which $f(\mathbb R)$ is not a regular curve. But if you consider a small open interval $U$ of $0\in\mathbb R$, the image is a small open piece of the vertical branch which is indeed a regular curve, but is not open in the lemniscate (it misses the horizontal branch).
This example stress the fact that an immersion is an embedding iff it is a homeo onto its image. This is important to understand and visualize projective plane and in general compact non-orientable surfaces. These surfaces can be embedded into $\mathbb R^4$ but not in $\mathbb R^3$. Since the next best thing to an embedding is an immersion, one looks at immersions into $\mathbb R^3$. For the projective plane there are several classical ones with famous names: the Steiner embeddings (like the roman surface or the crosscap), or the Boy surface. They have self-intersections and  singular points of course. A search in the web gives nice pictures of them all, as well as some of the standard immersion of the Klein Bottle in $\mathbb R^3$ (which justifies its name), which can even be seen in some beautiful glass models.
Another important class of immersions that are not embeddings are some "parametrizations" of manifolds, as the one of the sphere in the question. These are immersions whose image is in fact a smooth manifold. The are not homeos onto its image, but topological identifications. They give local parametrizations of that manifold when restricted to suitable open sets. Typical examples are the classical parametrizations of spheres, or tori.
A: When I first started learning about immersions it seemed weird to me that mathematicians would "go out of their way" to define them at all. There are two main reasons that it became important to define an immersed manifold, one is that they play a crucial role in Lie theory and the other is because they are needed to define a foliation of a manifold. When you start studying these things the usefulness of the definition of an immersed manifold will start to seem more natural.
But one of the main ways that I think about the difference between embedded and immersed submanifolds is that the topology of an embedded submanifold is the subspace topology, whereas the topology of an immersed manifold is in general finer than the subspace topology (has more open sets than the subspace topology).
