# Are open immersions equivalent to local diffeomorphisms? (not algebraic geometry)

Local diffeomorphisms are both open maps and immersions.

For the other direction:

Let $$N$$ and $$M$$ be smooth manifolds with dimensions. Let $$p \in N$$. Let $$F: N \to M$$ be a smooth map. Assume $$F$$ is an open map. Let $$F$$ be an immersion at p.

1. Because $$F$$ is an immersion at $$p$$, $$\dim N \le \dim M$$ at $$p$$ and thus at every $$q \in N$$ because $$N$$ and $$M$$ have dimensions.

2. Because $$F$$ is smooth and open, $$\dim N \ge \dim M$$, by this rule, from Momentum Maps and Hamiltonian Reduction By Juan-Pablo Ortega and Tudor Ratiu, which I now paraphrase:

Let $$N$$ and $$M$$ be smooth manifolds with dimensions. Let $$p \in N$$. Let $$F: N \to M$$ be a smooth map. If $$F$$ is open, then $$\dim N \ge \dim M$$.

(I know submersions are both open maps and imply $$\dim N \ge \dim M$$, but I didn't know openness is enough to imply $$\dim N \ge \dim M$$. Then again, I haven't studied this book. I just found some pages of it in a google search. I could be wrong in applying or understanding this rule since the definitions of manifold or smooth might be different)

1. By (1) and (2), $$\dim N = \dim M$$.

2. By (3) and this, $$F$$ is a local diffeomorphism at $$p$$ if and only if $$F$$ is an immersion at $$p$$ (if and only if $$F$$ is a submersion at $$p$$).

3. By (4), $$F$$ is a local diffeomorphism at $$p$$.

4. Therefore, by (5), open and immersion at $$p$$ implies local diffeomorphism at $$p$$.

5. I don't believe "openness" is defined pointwise. Therefore, by (6), open and immersion (everywhere) implies local diffeomorphism (everywhere)

• What is your question? Immersion means that the differential is everywhere an injective linear map. If the dimensions of $M, N$ agree then an immersion has an invertible differential at every point, by the inverse function theorem the map must then be a local diffeomorphism. Openness guarantees that the dimensions of $M$ and $N$ are equal by invariance of domain, but that is overkill as open is a much stronger result than that. – s.harp Jul 23 '19 at 8:55
• @s.harp Thanks. The question is in the title... "Are open immersions equivalent to local diffeomorphisms? (not algebraic geometry)" – user636532 Jul 23 '19 at 8:57

Having an open immersion guarantees that $$F$$ is a map between manifolds of equal dimension, and we know that immersions are equivalent to local diffeomorphisms in that context.