Are open local embeddings equivalent to local diffeomorphisms? (Do not use immersions)

Question

Open immersions are equivalent to local diffeomorphisms, and immersions are equivalent to local embeddings, so obviously yes. I would like to understand why open local embeddings equivalent to local diffeomorphisms without using immersions as equivalent to local embeddings.

1. This is how I understand local embeddings and local diffeomorphisms, and why I think open local embeddings equivalent to local diffeomorphisms. Is this correct?

   • Local diffeomorphism:

   For $X$ and $Y$ smooth manifolds with dimensions. A function $f:X\to Y$, is a local diffeomorphism, if for each point x in X, there exists an open set $U$ containing $x$, such that $f(U)$ is open in $Y$ and $f|_{U}:U\to f(U)$, is a diffeomorphism.

   • Local embedding:

   For $X$ and $Y$ smooth manifolds with dimensions. A function $f:X\to Y$, is a local embedding, if for each point x in X, there exists an open set $U$ containing $x$, such that $f(U)$ is a regular submanifold of $Y$ and $f|_{U}:U\to f(U)$, is a diffeomorphism.

   The only difference then is the codimension of $f(U)$ in both definitions. The codimension of (each) $f(U)$ is zero if and only if (each) $f(U)$ is open if and only if $f(X)$ is open if and only if $f$ is an open map.

2. This definition of local diffeomorphism as stated is wrong, if not some different definition, as talked about here because the definition is missing any of the 4 following equivalent conditions: $\dim N = \dim M$, $F$ is an open map, $F(N)$ is an open subset of $M$, or each $F(U)$ is open (Mindlack may have an issue with the last one! Haha). Yesterday I thought it could be a different definition, but then I realized that tangent spaces aren't introduced until 2 sections later, so I think this is indeed a mistake and not a different definition. (This is not exactly a problem in the book because whenever local diffeomorphisms are involved, we usually have an assumption of $\dim N = \dim M$.)

My question 2 initially was: What is the definition as stated a definition of then?

Initially, I thought it could define local embedding (equivalent to immersion, introduced 2 sections later; embedding is introduced 5 sections later) or local diffeomorphism onto image. After some thought, shown in the edits here, I change question 2 now:

My question 2 now is: Is the following correct?

• 2A. The definition as stated does not define local embedding, local diffeomorphism onto image or local diffeomorphism.

• 2B. (2A) is because the definition as stated does not describe the manifold structure of each of the $F(U)$'s.

• 2C. If each of the $F(U)$'s is open, then the definition is of local diffeomorphism.

• 2D. If each of the $F(U)$'s is open in $F(N)$, then the definition is of local diffeomorphism onto image, where $F(N)$ turns out to be a submanifold of $M$ (submanifold is not defined until 3 sections later) by this, which relies on this.

• 2E. If each of the $F(U)$'s is a submanifold of $M$ then the definition is of local embedding.

Answer 1

1. Okay for $f(U)$ has codimension $0$ iff it is open, not okay for the rest (given one $U$, you can’t know all of $f(X)$ being open if you only know $f(U)$ open).

A valid argument is that if $f$ is an open local embedding, then all the $f(U_x)$ for every $x$, are open, thus fulfilling the definition of diffeomorphism.

Conversely, if $f$ is a local diffeomorphism, then $f$ is a local embedding. Furthermore, if $V \subset X$, is an open subset of $X$, then for all $x \in V$, $f(V \subset U_x)$ is an open subset of $f(U_x)$ (thus of $Y$) containing $f(x)$, thus $f(V)$ is a neighborhood of $f(x)$. So $f(V)$ is open and $f$ is open. Therefore $f$ is an open local embedding.

2. I don’t understand your point. The idea of basic manifold theory is that we have a formalism that does not depend on dimensions or coordinates. The definition you are quoting (in the second link) is entirely correct. It does require $f(U)$ being open, of course.