Local Immersion Theorem

Question

I clearly understood the statement and proof of the theorem but the text gives few corollaries and remarks which i don't quite grasp such as

1. Immersion at x implies Immersion in a sufficiently small neighborhood.(I have no idea how it follows)

2. If both Manifolds are of same dimension then Immersions are just local diffeomorphisms( This one I got, Using the IFT)

3.Sufficiently small neighborhoods are mapped diffeomorphically onto its range under Immersion.( It's quite obvious from 2 but it's remarked after the Theorem How??)

Kindly help!!

Thanks & regards

Answer 1

Let $M$ and $N$ be smooth manifolds of dimension $m$ and $n$ respectively and let $f:M \to N$ be a smooth map which is an immersion at a point $x\in M$. Then, by the local immersion theorem we can put it into a particularly nice canonical form: there is a chart $(U,\alpha)$ of $M$ , and there is a chart $(V,\beta)$ of $N$ such that

• $x \in U$, and $\alpha(x) = 0$, and $f(U) \subseteq V$
• $\beta \circ f\big|_U \circ \alpha^{-1} = \iota\big|_{\alpha[U]}$, where $\iota :\Bbb{R}^m \to \Bbb{R}^m \times \Bbb{R}^{n-m}$, $x \mapsto (x,0)$ is the canonical immersion.

This is a very useful statement, because if you rewrite the above compositions, we have that \begin{align} f\big|_{U} = \beta^{-1} \circ \iota\big|_{\alpha[U]} \circ \alpha \end{align} Now, note that $\iota$ is clearly an immersion (it's an injective linear map), and that since $\alpha$ and $\beta$ are diffeomorphisms, they're trivially immersions as well. Therefore $f\big|_U$ is the composition of three immersions, and hence is an immersion as well. Thus, we have shown that $f$ is an immersion in a neighbourhood of the point $x$.

Lastly, for proving $(3)$, once again make use of these same charts $(U,\alpha)$ and $(V,\beta)$ to prove that $f(U)$ is a submanifold of $N$ (simply follow the definitions). Next, you need to show that $f\big|_U$ is injective (this should be easy from the above discussion), and hence bijective onto its image. Lastly, it shouldn't be too hard to prove that the inverse $(f\big|_U)^{-1}$ is smooth; you simply have to apply the definition of smoothness and check that it works.

The key to this entire problem is the existence of the special charts $(U,\alpha)$ and $(V,\beta)$ which make $f$ "look like" the canonical immersion $\iota$. So if at any point along the way you get stuck, just ask yourself: "did I make full use of these special charts?"