# Are these definitions of (what might be) “local diffeomorphism onto image” equivalent?

Update: By this, I think these definitions are indeed equivalent.

The following definitions might not be what everyone means by "local diffeomorphism onto image" because there may be different conventions because some might consider immersions to be "local diffeomorphisms onto image". For the sake of having a title, I just consider these definitions to be for what one might call "local diffeomorphism onto image". My question is: Are these equivalent? Please verify.

• Convention: For this post, submanifold means regular/embedded submanifold.

• Note 1: The following definitions are of "local diffeomorphism onto image" everywhere rather than for pointwise to accommodate that Definition 3 isn't explicitly pointwise. I'll also define local diffeomorphism everywhere.

For Definitions 1, 2 and 3. Let $$M$$ and $$N$$ be smooth manifolds with dimensions. Let $$F: N \to M$$ be a smooth map. Its image is $$F(N)$$. Let $$i: F(N) \to M$$ be the inclusion map, which may or may not be smooth. Let $$G=\tilde F: N \to F(N)$$ be $$F$$ with restricted range that satisfies $$F = \iota \circ G = \iota \circ \tilde F$$. $$G$$ also may or may not be smooth.

• Convention: For this post, $$\tilde H$$ means restricting the range a map $$H$$ to its image.

Definition of local diffeomorphism. First, we define local diffeomorphism: We say $$F$$ is a local diffeomorphism if for all $$p \in N$$, there exists an open neighborhood $$V_p$$ of $$p$$ in $$N$$ such that $$F(V_p)$$ is open in $$M$$ and that $$\tilde{F|_{V_p}}: V_p \to F(V_p)$$ is a diffeomorphism (Given that $$F(V_p)$$ is a submanifold of $$M$$, even if $$F(V_p)$$ were not open, we have that $$\tilde{F|_{V_p}}$$ is a diffeomorphism is equivalent to that $$F|_{V_p}: V_p \to M$$ is an embedding).

Now, let us say $$F$$ is a local diffeomorphism onto its image if

Definition 1. $$F(N)$$ is a submanifold of $$M$$, and $$G$$ is a local diffeomorphism.

Definition 2. For all $$p \in N$$, there exists an open neighborhood $$U_p$$ of $$p$$ in $$N$$ such that $$F|_{U_p}: U_p \to M$$ is an embedding and $$F(U_p)$$ is open in $$F(N)$$.

• This is my understanding of Definition 3 below, which is from Arrow.

• Note 2: I believe Definition 2 implies rather than assumes that $$F(U_p)$$ is a submanifold of $$M$$, depending on your definition of embedding (or maybe even not depending).

• Note 3: This is like how we can define local diffeomorphism as either of manifolds with dimension of the same dimension without assuming that $$F(V_p)$$ open because we would just deduce $$F(V_p)$$ open anyway (which I didn't do above) or of manifolds with dimension of unspecified dimension that says $$F(V_p)$$ open because we would just deduce the manifolds have the same dimension anyway (which I did above).

Definition 3. There exists an open cover $$\{W_i\}_{i \in I}$$ of $$N$$ such that $$F|_{W_i} \to M$$ is an embedding and $$F(W_i)$$ is open in $$F(N)$$.

• This is from Arrow.

• Note 4: I believe Definition 3 implies rather than assumes that $$F(W_i)$$ is a submanifold of $$M$$. See Notes 2 and 3.

A. Definition 1 implies Definition 2:

Choose $$U_p = V_p$$. We must show that

1. Ai. $$F(U_p)$$ is a submanifold of $$M$$ (first part of showing $$F|_{U_p}$$ is an embedding).

2. Aii. $$\tilde{F|_{U_p}}: F(U_p) \to F(U_p)$$ is a diffeomorphism (second part of showing $$F|_{U_p}$$ is an embedding).

3. Aiii. $$F(U_p)$$ is open in $$F(N)$$.

This works because:

1. A1. Submanifold is a transitive property: For $$A \subseteq B \subseteq C$$, if $$A$$ is a submanifold of $$B$$, and $$B$$ is of $$C$$, then $$A$$ is of $$C$$. I think. (I don't know how to prove this directly for regular submanifold, but I think I know how to prove that embedded submanifold is a transitive property).

2. A2. Therefore, we have (Aiii): $$F(U_p)=G(V_p)$$ is open in $$G(N)=F(N)$$ because $$G$$ is a local diffeomorphism.

3. A3. $$F(U_p)=G(V_p)$$ is a submanifold of $$F(N)$$ (with codimension zero) by (A2).

4. A4. Therefore, we have (Ai): $$F(U_p)=G(V_p)$$ is a submanifold of $$M$$ by (A3) and (A1).

5. A5. Therefore, we have (Aii): By (A2) or (A4), it makes sense to discuss whether or not $$\tilde{F|_{U_p}}$$ a diffeomorphism. $$\tilde{F|_{U_p}}$$ is a diffeomorphism since $$\tilde{F|_{U_p}}$$ is identical to $$\tilde{G|_{V_p}}$$.

B. Definition 2 implies Definition 1:

Choose $$V_p=U_p$$. We must show that

1. Bi. $$F(N)=G(N)$$ is a submanifold of $$M$$.

2. Bii. $$G(V_p)$$ is open in $$G(N)=F(N)$$

3. Biii. $$\tilde{G|_{V_p}}: V_p \to G(V_p)$$ is a diffeomorphism, assuming $$G(V_p)$$ is a submanifold of $$M$$ (I think we cannot assume only that $$G(V_p)$$ is a manifold subset of $$M$$).

This works because:

1. B1. Whether or not $$F(N)=G(N)$$ is a submanifold of $$M$$, $$F(N)$$ surely is a topological subspace of $$M$$, so it makes sense to talk about whether or not $$G(V_p)$$ is open in $$G(N)=F(N)$$.

2. B2. Therefore, we have (Bii): $$G(V_p)=F(U_p)$$ is open in $$G(N)=F(N)$$, by (B1).

3. B3. Therefore, we have (Bii): Assuming (Bi) holds, we have by (B1) and (A1) that $$G(V_p)$$ is a submanifold of $$M$$, so it makes sense to discuss whether or not $$\tilde{G|_{V_p}}$$ a diffeomorphism. $$\tilde{G|_{V_p}}$$ is a diffeomorphism since $$\tilde{G|_{V_p}}$$ is identical to $$\tilde{F|_{U_p}}$$.

4. B4. By this, (Bi) holds if (and only if, I think) for all open subsets $$P$$ of $$N$$, $$F(P)=G(P)$$ is open in $$F(N)=G(N)$$.

5. B5. Let $$P$$ be open in $$N$$. Then for all $$p \in P$$, since $$p \in N$$, there exists a neighborhood $$Q_p$$ of $$p$$ in $$N$$ such that $$F(Q_p)$$ is open in $$F(N)$$, $$F(Q_p)$$ is a submanifold of $$M$$ and $$\tilde{F|_{Q_p}}: Q_p \to F(Q_p)$$ is a diffeomorphism.

6. B6. By (B5), pick $$Q_p$$'s for each $$p \in P$$ to get $$F(P)=F(P \cap [\bigcup_{p \in P} Q_p]) = F(\bigcup_{p \in P} [P \cap Q_p]) = \bigcup_{p \in P} F([P \cap Q_p])$$

7. B7. The $$P \cap Q_p$$'s in (B6) are open in $$Q_p$$.

8. B8. The $$F([P \cap Q_p])$$'s in (B6) are open in $$F(Q_p)$$, by (B5) and (B7).

9. B9. $$F(P)$$ is open in $$F(N)$$, by (B8)

10. B10. Therefore, we have (Bi): $$F(N)=G(N)$$ is a submanifold of $$M$$ by (B9) and (B4).

C. Definition 2 implies Definition 3

Pick at least 1 $$U_p$$ for each $$p$$ and then collect them all together. We index by $$i \in I$$ rather than $$p \in N$$ to indicate that we may pick more than one $$U_p$$ for each $$p$$.

D. Definition 3 implies Definition 2

For each $$p \in U_p$$, choose $$U_p$$ to be any $$W_i$$ such that $$p \in W_i$$.