Definition and clarification of submanifold, immersion and submersion

Question

I'm learning Differential geometry by myself and I got stuck with the definition of immersion.

My questions were:

1. Could you give me the definition of submanifold, immersion and submersion? and what's their connection?

2. I had a hard time understanding immersion, I'm not quiet sure what the wiki was saying https://en.wikipedia.org/wiki/Immersion_(mathematics) . Is immersion a function with respect to $p$ or is it a function with respect to the neighberhood of $p$? (Is $p$ a fixed point?)

Further, I'm reading Differential Geometry: Bundles, Connections, Metrics and Curvature by Clifford Henry Taubes where in pages 7, he gave a much simplier definition of immersions:

A smooth map $\psi:Y\rightarrow M$ was said to be immersion when fix any point $p\in Y$ and open set $V\subset Y$that contain $p$ with map $\phi_V:V\rightarrow R^n$, and $U\in M$ denote an open set containing $\psi(p)$ with a coordinate map $\phi_U:U\rightarrow R^m$. then the differential of $\phi_u\circ \psi\circ\phi_V^{-1}$ at $\phi_V(p)$ was an injective linear map from $R^n$ to $R^m$.

3. What does "differential of $\phi_u\circ \psi\circ\phi_V^{-1}$ at $\phi_V(p)$ " mean? I wasn't quiet sure how to take the differential of this function. Do we need to establish a direction $a$ or what?

4. He also said that "This implies that $\psi(V)$ is an n dimesional submanifold of M on some small neighborhood of $\psi(p)$." How does this relate to the second definition in wiki that "rank $D_pf=\dim M$"?

5. Also,a further request, could you give me a formal definition of coordinate map that was used in defining the definition?

Answer 1

1. There are two notions (that I am aware) of submanifolds. I will write that once you understand about immersion and submersion. A smooth map $F:M\rightarrow N$ is called an immersion (respectively submersion) if for each $p\in M$, differential of $F$ at $p$ i.e., $F_{*,p}:T_pM\rightarrow T_{F(p)}N$ is an injective (respectively surjective) map.
2. You said "Is immersion a function with respect to $p$ or is it a function with respect to the neighberhood of $p$? (Is p a fixed point?)".. Definition of immersion (respectively submersion) says that some condition on differential holds for each $p$.
3. You said "What does "differential of $\phi_u\circ \psi\circ\phi_V^{-1}$ at $\phi_V(p)$ " mean? I wasn't quiet sure how to take the differential of this function. Do we need to establish a direction $a$ or what?" This is just the differential of $F$ at $p$. Do you know what is differential of a smooth function between manifolds?

This is what I would like to say for now. If you understand this clearly, I would try explaining other things. Otherwise, there is no point in writing everything.