Let $N$ and $M$ be smooth manifolds of respective dimensions $n$ and $m$. Let $F:N \to M$ be a smooth map.
Please verify my proof of the equivalence of the following 2 definitions.
From An Introduction to Manifolds by Loring W. Tu: Definition 1: Immersion and topological embedding
From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave: Definition 2: Image is smooth (regular) submanifold (and thus manifold) and diffeomorphism onto image
To prove Definition 1 implies Definition 2:
Image is smooth submanifold: Tu Theorem 11.13
Diffeomorphism onto image: Let $i: F(N) \to M$ be inclusion. Then the restriction $\tilde F: N \to F(N)$, which satisfies $F = i \circ \tilde F$ is smooth since $F$ smooth by Tu Theorem 11.15. $\tilde F$ is also a diffeomorphism through these steps:
Step 1: $\tilde F$, like the original $F$, is an immersion.
Step 2: $N$ and $F(N)$ have the same dimension.
Step 3: $\tilde F$ is a local diffeomorphism, i.e. $F$ is a local diffeomorphism onto its image.
- This follows by Steps 1 and 2 because immersions of manifolds of the same dimension are local diffeomorphisms, as proven here.
Step 4: $\tilde F$ is a diffeomorphism
- This follows by Step 3 because any smooth map $G$ of manifolds (with dimensions) is a diffeomorphism if and only if $G$ is a bijective local diffeomorphism, as proven here and here.
To prove Definition 2 implies Definition 1:
Homeomorphism onto image: Diffeomorphism onto image implies homeomorphism onto image, i.e. $\tilde F$ diffeomorphism implies $\tilde F$ homeomorphism.
Immersion: Diffeomorphism onto image implies $\tilde F$ is immersion. Then, $F$ is also an immersion by this again: $\tilde F$ immersion is equivalent to $F$ immersion