intuition for definition of immersions

Question

A smooth map $f: M \to N$ is said to be an immersion if $df_p$ is injective for all $p\in M$.I have a hard time in interpreting the meaning of injectivity in this definition. Thanks in advance.

Answer 1

I think the key insight about immersions comes from the Rank Theorem: If $f: M\to N$ is an immersion, then for each $p\in M$ there are coordinates $(x^1,\dots,x^m)$ on a neighborhood of $p$ and coordinates $(y^1,\dots,y^m,y^{m+1},\dots,y^n)$ on a neighborhood of $f(p)$ such that the local coordinate representation of $f$ is given by $$ f(x^1,\dots,x^m) = (x^1,\dots,x^m,0,\dots,0).\tag{$\ast$} $$ In other words, up to a change of coordinates, an immersion looks locally like the natural inclusion of $\mathbb R^m$ into $\mathbb R^n$.

On the other hand, any map that has a representation in local coordinates as inclusion of a linear subspace automatically has injective differential. So the real reason for the definition is this: a smooth map has injective differential everywhere if and only if in a neighborhood of each point it has a coordinate representation which is inclusion of a linear subspace.