What's the lemma's name ?And how to proof it?

Question

I saw the lemma in a Riemannian Geometry book, but the proof is omitted ,I want to know how to proof it .Thanks.

Answer 1

There is a theorem from elementary differential topology known as the Rank Theorem, or the Constant Rank Theorem. Here is the statement:

Let $M, N$ be manifolds of dimension $m, n$ respectively. Let $f: M\to N$ be a smooth map of constant rank $r$. Then for each point $p\in M$, there are coordinate charts centered at $p$ and $f(p)$ in which the coordinate representation of $f$ is $$f(x^1, \ldots, x^r, x^{r+1},\ldots x^m) = (x^1,\ldots,x^r,0,\ldots,0)$$

In particular, an immersion is locally of the form $f(x^1,\ldots,x^m) = (x^1,\ldots,x^m,0,\ldots,0)$. The proof of this theorem is a bit involved so I won't reproduce it here. It can be found in many books on differential topology/geometry.

By the coordinate representation, $f$ is an injective immersion when restricted to the chosen coordinate chart $\tilde{U}$. To promote it to an embedding, we can choose a precompact open ball $U$ centered at $p$ such that $\overline U \subset \tilde U$. Then $f|_{\overline U}$ is an injective continuous map whose domain is compact, hence it is a homeomorphism onto its image by the closed map lemma. It follows that $f|_U$ is an embedding.