# Proof verification that bijective local diffeomorphisms are diffeomorphisms

My book is An Introduction to Manifolds by Loring W. Tu.

I think that since diffeomorphisms are equivalent to surjective smooth embeddings, I think this is the same as proving an injective local diffeomorphism is a smooth embedding (I understand "image is submanifold and diffeomorphism onto its image" is an equivalent definition of a smooth embedding) which is asked here and here (and here in the continuous case), but I would like a verification of my own proof:

1. Let $$M$$ and $$N$$ be smooth manifolds of respective dimensions $$m$$ and $$n$$. Let $$F: N \to M=F(N)$$ be a bijective smooth map. Suppose $$F$$ is a local diffeomorphism. Let us show that $$F^{-1}: M \to N$$ is smooth, using Definition 6.5, to show that $$F$$ is actually a diffeomorphism.

2. Let $$U$$ be open in $$N$$. I'll denote by

• $$F|_U$$ as the domain-restriction $$F|_U:U \to M$$,

• $$\tilde{F|_U}$$ as the range-restriction $$\tilde{F|_U}:U \to F(U)$$

• $$G$$ as the inverse of $$\tilde{F|_U}$$, i.e. $$G := (\tilde{F|_U})^{-1}: F(U) \to U$$.

• Equivalently, $$G = \tilde{F^{-1}|_{F(U)}}: F(U) \to U$$, the range-restriction of the domain-restriction $$F^{-1}|_{F(U)}: F(U) \to N$$ of $$F^{-1}: M \to N$$
3. I understand that the "$$\varphi \circ F^{-1} \circ \psi^{-1}$$" in Definition 6.5 refers to "$$\varphi \circ G \circ \psi^{-1}$$"

4. For all $$F(p) \in M$$, with $$p \in N$$, we must find a chart $$(U, \varphi)$$ about $$p$$ in $$N$$ and a chart $$(V, \psi)$$ about $$F(p)$$ in $$M$$ such that $$\varphi \circ G \circ \psi^{-1}: \psi(F(U) \cap V) \to (\varphi \circ G)(F(U) \cap V)$$ is smooth about the point $$\psi(F(p))$$ (its value at $$\psi(F(p))$$ is $$\varphi(p)$$).

5. Let us now use the local diffeomorphism property: For all $$p \in N$$, there exists a neighborhood $$U_p$$ of $$p$$ in $$N$$ such that for the domain-restriction $$F|_{U_p}: U_p \to M$$ and the range-restriction $$\tilde{F|_{U_p}}: U_p \to F(U_p)$$, $$F(U_p)$$ is open in $$M$$ and $$\tilde{F|_{U_p}}$$ is a diffeomorphism.

6. I'm not sure if $$U_p$$ has a coordinate map $$\gamma$$ that makes $$(U_p,\gamma)$$ into a chart, but since $$U_p$$ is an open subset of $$N$$, $$U_p$$ can be made into a smooth manifold. Viewing $$p \in U_p$$, there exists a chart $$(A_p,\varphi_A)$$ about $$p$$ in $$U_p$$. I know $$A_p$$ is open in both $$U_p$$ and $$N$$, and I think $$(A_p,\varphi_A)$$ is also a chart about $$p$$ in $$N$$. (If relevant: I guess we use that open subsets are equivalent to regular submanifolds of codimension zero.)

7. $$n=m$$ because

• $$n = \dim N = \dim U_p = \dim F(U_p) = \dim M = m$$, where

• $$\dim F(U_p)$$ is defined because $$F(U_p)$$ becomes a manifold because $$F(U_p)$$ is an open subset of $$M$$ (from (5))

• $$\dim U_p = \dim F(U_p)$$ because of (3) in another post

8. I'll choose $$(U, \varphi) = (A_p,\varphi_A)$$ and $$(V, \psi) = (G^{-1}(A_p), \varphi_A \circ G)$$. This works because

• 8.1 $$\varphi \circ F^{-1} \circ \psi^{-1} = \varphi \circ G \circ \psi^{-1} = \varphi_A \circ G \circ (\varphi_A \circ G)^{-1}$$

$$= \varphi_A \circ G \circ G^{-1} \circ \varphi_A^{-1} = \varphi_A \circ \varphi_A^{-1}$$

• 8.2 $$\psi(F(U) \cap V) = (\varphi_A \circ G)(F(A_p) \cap G^{-1}(A_p)) = (\varphi \circ G)(F(U) \cap V)$$ (which turns out to be equal to $$\varphi_A(A_p)$$)

• 8.3 By (8.2) and (7), it makes sense to say that (8.1) shows that $$\varphi \circ F^{-1} \circ \psi^{-1}$$ is an identity map on an open subset of $$\mathbb R^m$$.

9. Identity maps on open subsets of $$\mathbb R^m$$ are smooth.

10. Therefore, by (8) and (9), $$F^{-1}$$ is smooth.

11. Therefore, by (10) and (1), $$F$$ is a diffeomorphism.

• Try using the Proof Verification tag, you might get more attention Jul 20, 2019 at 12:28
• @theREALyumdub THANKS!
– user636532
Jul 20, 2019 at 12:29

Your proof is fine. But unless I am missing something, or using a different definition of "locally smooth", I would simply argue as follows: $$F:M\to N$$ is bijective, so it has an inverse $$F^{-1}:N\to M$$. Since $$F$$ is a local diffeomorphism, for each $$p\in M,\ F^{-1}$$ is differentiable at $$F(p).$$ It follows that $$F$$ is a diffeomorphism.
• Thanks. I assume you mean by "differentiable" you mean "smooth". How do you prove that "Since $F$ is a local diffeomorphism, for each $p\in M,\ F^{-1}$ is differentiable (smooth) at $F(p)$" ? I think I ended up proving something like that here and in doing so relied on this very question. Something's up. It seems that something that would be circular actually isn't (well, I hope so).
• Yes, differentiable= smooth, i.e. $\psi\circ F\circ \phi^{-1}$ is differentiable for appropriately chosen charts. $F^{-1}$ is smooth $by definition$ of diffeomorphism. The point is, differentiability is a local property to begin with, so the result is more or less automatic, once you have that additionally, $F$ is bijective. Jul 21, 2019 at 17:31
• Thanks. Actually 1. Is this true? "A map $G: P \to Q$ of smooth manifolds is smooth if each $r \in P$ has a neighborhood $V_r$ in $P$ such that $G: V_r \to Q$ is smooth." I seem to have written it in another post but am not sure where I got that. This sounds like what you meant by "differentiability is a local property". (That differentiability is local is a concept is somewhere in Section 19 I think, but I can't find exactly anything like the quoted sentence)