$F:M\to N$ is a local diffeomorphism iff it is both a smooth immersion and a smooth submersion.

Question

Suppose $M$ and $N$ are smooth manifolds without boundary. I'd like to show that $F:M\to N$ is both a smooth immersion and a smooth submersion whenever $F$ is a local diffeomorphism. Fix $p\in M$. By employing the definition of a local diffeomorphism, John M. Lee, the author of my textbook, claims that there exists a neighborhood $U$ of $p$ s.t. $F$ maps $U$ diffeomorphically onto $F(U)$. From this, he concludes that $dF_p:T_p M\to T_{F(p)}N$ is an isomorphism. This confuses me a lot. As far as I know, one can say so when $F$ is globally a diffeomorphism. But, in our case, $F$ is merely a local diffeomorphism. What's the argument for this, please? BTW, I haven't come to the chapter about submanifolds yet. Thank you.

Answer 1

This is one pedantic way to see it. Let $F(p) = q$ and $V=F(U)$. Suppose that $f: U \to V$ is the diffeomorphism obtained from the restriction of $F$ to open subsets $U \subseteq M$ and $V \subseteq N$.

We know that $df_p : T_pU \to T_{q}V$ is an isomorphism. We can relate $df_p$ to $dF_p$ through isomorphisms $di_p : T_pU \to T_pM$ and $dj_{q} : T_{q}V \to T_qN$, where $i : U \hookrightarrow M$ and $j : V \hookrightarrow N$ are the inclusion maps. We observe that $$\require{AMScd} \begin{CD} U @>{f}>> V\\ @V{i}VV @VV{j}V \\ M @>{F}>> N \end{CD}, \quad \text{that is} \quad j \circ f = F \circ i.$$ So the differentials related as $dj_q \circ df_p = dF_p \circ di_p$. All three of them known to be isomorphisms, so $dF_p$ is also an isomorphism. Another way is to compute directly the Jacobian matrix of the differentials on a chart, which is somewhat simpler.