# An injective immersion is an embedding if and only if open onto image if and only if image is (regular) submanifold?

Update: Tim kinsella appears to say yes here.

Related:

1. An injective smooth function with injective differential must have a continuous inverse?

2. I think my question is equivalent to the converse of this.

Let $$N$$ and $$M$$ be smooth manifolds with dimensions. Let $$p \in N$$. Let $$F: N \to M$$ be a smooth map.

A. $$F$$ is said to be an embedding if $$F$$ is an immersion and a topological embedding. (You may have a different but equivalent definition.)

B. Let $$X$$ and $$Y$$ be topological spaces. A map, not necessarily injective or continuous, $$F: X \to Y$$ is said to be a topological embedding if $$\tilde F: X \to F(X)$$ is a homemorphism.

C. $$F$$ is said to be open onto its image if $$\tilde F$$ is open.

D. If $$F$$ is an injective immersion, then $$\tilde F$$ is already injective ($$\tilde F$$ is injective if and only if $$F$$ is too), surjective and continuous (because $$F$$ is continuous because $$F$$ is smooth). Therefore, $$F$$ is an embedding if and only if $$F$$ is open onto its image.

Question: Is $$\tilde F$$ open if and only if $$F$$ is an embedding if and only if $$F(N)$$ is a regular submanifold (with dimension) of $$M$$? I think yes. Please verify.

Suppose $$\tilde F$$ open. Show $$F(N)$$ is a regular submanifold of $$M$$: Done here (which assumes open onto image rather than open).

Suppose $$F(N)$$ is a regular submanifold of $$M$$. Show $$\tilde F$$ is open.

1. Because $$F(N)$$ is a regular submanifold of $$M$$, $$F(N)$$ is a manifold, so $$\tilde F$$ is a map of manifolds, so we can talk about $$\tilde F$$ as smooth or not smooth. Unsurprisingly, because $$F(N)$$ is a regular submanifold of $$M$$, $$\tilde F$$ is smooth by this. Thus, we can talk about $$\tilde F$$ possibly being an immersion, submersion, local diffeomorphism, embedding, diffeomorphism, etc.

2. $$\tilde F$$ is an immersion if and only if $$F$$ is an immersion, by (1) and this,

3. $$\tilde F$$ is a surjective immersion, by (2) and (D),

4. $$\dim F(N) = \dim N$$, by (3) and this (or by (3) and this, I guess; Maybe the two arguments, one of using this and one of using this are actually equivalent).

5. $$\tilde F$$ is both a (surjective) submersion and a (surjective) local diffeomorphism, by (4) and this.

6. Both submersions and local diffeomorphisms are open maps.

7. By (5) and (6), $$\tilde F$$ is open.

$$\tilde{F}: N \rightarrow F(N)$$ is a smooth bijection with non degenerate derivative between two manifolds. so it's a diffeomorphism. in particular, it's an open map.